Class 6 Mathematics – NCERT (Ganita Prakash)
Chapter 8: PLAYING WITH CONSTRUCTIONS - Complete Question Bank
A. Multiple Choice Questions (20 Questions)
1. In a rectangle, how many pairs of opposite sides are equal?
Answer: b) 2
A rectangle has two pairs of opposite sides: the two lengths and the two breadths.
(Logical Reasoning, Spatial Understanding)
2. If all sides of a quadrilateral are equal and all angles are 90°, it is a:
Answer: c) Square
A square is defined by exactly these two properties: all sides equal and all angles 90°.
(Conceptual Understanding)
3. How many right angles does a square have?
Answer: d) 4
By definition, every angle in a square is a right angle (90°).
(Factual Recall)
4. Which of the following is NOT a valid name for a rectangle with vertices A, B, C, D?
Answer: c) ACBD
Vertices must be named in order around the shape. ACBD jumps from A to C (a diagonal).
(Spatial Understanding)
5. Which property is true for both squares and rectangles?
Answer: c) All angles 90°
Both have four right angles. Only squares have all sides equal.
(Analytical Thinking)
6. If a diagonal of a rectangle divides an angle into 60° and 30°, the other angles are:
Answer: a) 60° and 30°
Diagonals create equal angles at opposite vertices.
(Logical Reasoning)
7. To construct a square of side 5 cm, which step comes first?
Answer: b) Draw a line of 5 cm
First draw one side as the base.
(Procedural Understanding)
8. A rectangle with sides 4 cm and 6 cm has a perimeter of:
Answer: b) 20 cm
Perimeter = 2 × (4 + 6) = 20 cm.
(Numerical Application)
9. Which shape can be divided into two identical squares?
Answer: b) Rectangle with sides in ratio 1:2
Length must be exactly twice the breadth.
(Analytical Thinking)
10. How many arcs are drawn to locate a point equidistant from two given points?
Answer: b) 2
One arc from each point with the same radius.
(Procedural Understanding)
11. In a square, each diagonal divides the opposite angles into:
Answer: a) Two equal parts
Diagonals of a square bisect the angles (45° each).
(Conceptual Understanding)
12. If AB = 8 cm in a rectangle, then CD =
Answer: b) 8 cm
Opposite sides are equal in a rectangle.
(Logical Reasoning)
13. A rotated square is still a square because:
Answer: c) Both sides and angles remain same
Rotation doesn't change dimensions or angles.
(Spatial Understanding)
14. Which instrument is essential for constructing a "House" figure with arcs?
Answer: b) Compass
The arcs in the house construction require a compass.
(Tool Knowledge)
15. A rectangle with one side 5 cm and diagonal 7 cm will have the other side approximately:
Answer: a) 4.9 cm
Using Pythagoras: √(7² - 5²) = √24 ≈ 4.9 cm.
(Numerical Application)
16. The number of diagonals in a rectangle is:
Answer: b) 2
Every quadrilateral has 2 diagonals.
(Factual Recall)
17. What is the main use of a compass in geometric constructions?
Answer: b) To draw arcs and transfer lengths
Compass is primarily for arcs and transferring equal distances.
(Tool Knowledge)
18. In the "Wavy Wave" pattern, if the central line is 8 cm, what is the radius for half-circle waves?
Answer: b) 4 cm
For half-circle waves, radius = half of central line segment = 4 cm.
(Application)
19. When constructing a rectangle given one side and diagonal, we use:
Answer: b) Arcs from compass
We draw an arc with diagonal as radius to locate the third vertex.
(Procedural Understanding)
20. What do we call lines drawn lightly to guide construction?
Answer: b) Construction lines
These are temporary guiding lines erased later.
(Terminology)
B. Assertion & Reasoning (20 Questions)
1. Assertion (A): All squares are rectangles.
Reason (R): A square has all angles 90° and opposite sides equal.
Reason (R): A square has all angles 90° and opposite sides equal.
Answer: a) Both A and R are true, and R is the correct explanation of A.
A square satisfies both properties of a rectangle (all angles 90°, opposite sides equal).
(Logical Reasoning, Analytical Thinking)
2. Assertion (A): A rotated square remains a square.
Reason (R): Rotation changes the side lengths.
Reason (R): Rotation changes the side lengths.
Answer: c) A is true, but R is false.
Rotation preserves side lengths and angles.
(Spatial Understanding)
3. Assertion (A): Diagonals of a rectangle are equal.
Reason (R): Diagonals of a square are not equal.
Reason (R): Diagonals of a square are not equal.
Answer: c) A is true, but R is false.
Square diagonals are also equal.
(Conceptual Understanding)
4. Assertion (A): A rectangle can be named in 8 different ways.
Reason (R): The vertices can be taken in any order.
Reason (R): The vertices can be taken in any order.
Answer: c) A is true, but R is false.
Vertices must be in cyclic order, not any order.
(Analytical Thinking)
5. Assertion (A): To find a point equidistant from two points, we draw two arcs.
Reason (R): The intersection of arcs gives points at equal distance from both centers.
Reason (R): The intersection of arcs gives points at equal distance from both centers.
Answer: a) Both A and R are true, and R is the correct explanation of A.
The construction method follows from the geometric principle stated in R.
(Procedural Understanding)
6. Assertion (A): A square can be divided into two identical rectangles.
Reason (R): A square has all sides equal.
Reason (R): A square has all sides equal.
Answer: b) Both A and R are true, but R is not the correct explanation of A.
Both true, but division into rectangles is due to symmetry, not just equal sides.
(Analytical Thinking)
7. Assertion (A): In a rectangle, if one diagonal divides an angle into 45° and 45°, the rectangle is a square.
Reason (R): In a square, diagonals bisect the angles equally.
Reason (R): In a square, diagonals bisect the angles equally.
Answer: a) Both A and R are true, and R is the correct explanation of A.
If diagonal bisects 90° into 45°, adjacent sides must be equal, making it a square.
(Logical Reasoning)
8. Assertion (A): Using a compass, we can transfer lengths without a ruler.
Reason (R): A compass can measure angles.
Reason (R): A compass can measure angles.
Answer: c) A is true, but R is false.
Compass transfers lengths, not measures angles.
(Tool Knowledge)
9. Assertion (A): A rectangle with sides 3 cm and 4 cm has a diagonal of 5 cm.
Reason (R): For a rectangle, diagonal = √(length² + breadth²).
Reason (R): For a rectangle, diagonal = √(length² + breadth²).
Answer: a) Both A and R are true, and R is the correct explanation of A.
√(3² + 4²) = √(9+16) = √25 = 5 cm.
(Numerical Application)
10. Assertion (A): In the "House" construction, point A is found using two arcs.
Reason (R): Point A is 5 cm from both B and C.
Reason (R): Point A is 5 cm from both B and C.
Answer: a) Both A and R are true, and R is the correct explanation of A.
Because A is 5 cm from both B and C, we use arcs of radius 5 cm from both points.
(Application)
11. Assertion (A): A quadrilateral with all sides equal is a square.
Reason (R): A rhombus also has all sides equal.
Reason (R): A rhombus also has all sides equal.
Answer: d) A is false, but R is true.
All sides equal makes it a rhombus. Need all angles 90° for square.
(Conceptual Understanding)
12. Assertion (A): A rectangle cannot be divided into 3 identical squares if sides are not in ratio 1:3.
Reason (R): For 3 identical squares, longer side must be 3 times shorter side.
Reason (R): For 3 identical squares, longer side must be 3 times shorter side.
Answer: a) Both A and R are true, and R is the correct explanation of A.
The condition in R explains why A is true.
(Analytical Thinking)
13. Assertion (A): A square satisfies the following two properties: S1) All the sides are equal, and S2) All the angles are 90°.
Reason (R): These are the defining properties of a square.
Reason (R): These are the defining properties of a square.
Answer: a) Both A and R are true, and R is the correct explanation of A.
R identifies these as definitional properties.
(Conceptual Understanding)
14. Assertion (A): A square of side 6 cm has a diagonal of about 8.5 cm.
Reason (R): Diagonal of a square = side × √2.
Reason (R): Diagonal of a square = side × √2.
Answer: a) Both A and R are true, and R is the correct explanation of A.
6 × √2 ≈ 6 × 1.414 = 8.484 ≈ 8.5 cm.
(Numerical Application)
15. Assertion (A): In a rectangle, diagonals bisect each other.
Reason (R): Diagonals of a rectangle are perpendicular.
Reason (R): Diagonals of a rectangle are perpendicular.
Answer: c) A is true, but R is false.
Rectangle diagonals bisect but are perpendicular only in square.
(Conceptual Understanding)
16. Assertion (A): To construct a rectangle given one side and a diagonal, we use a compass.
Reason (R): The third vertex lies on the intersection of an arc and a perpendicular line.
Reason (R): The third vertex lies on the intersection of an arc and a perpendicular line.
Answer: a) Both A and R are true, and R is the correct explanation of A.
The arc (from compass) with diagonal as radius helps locate the vertex.
(Procedural Understanding)
17. Assertion (A): A rotated rectangle is still a rectangle.
Reason (R): Rotation does not change lengths and angles.
Reason (R): Rotation does not change lengths and angles.
Answer: a) Both A and R are true, and R is the correct explanation of A.
Since rectangle properties depend on lengths/angles preserved by rotation.
(Spatial Understanding)
18. Assertion (A): A compass can be used to draw a circle of radius 4 cm.
Reason (R): A compass has a pencil and a pointed tip.
Reason (R): A compass has a pencil and a pointed tip.
Answer: b) Both A and R are true, but R is not the correct explanation of A.
R describes structure, not how radius is set and maintained.
(Tool Knowledge)
19. Assertion (A): A rhombus with all angles 90° is a square.
Reason (R): A rhombus has all sides equal.
Reason (R): A rhombus has all sides equal.
Answer: a) Both A and R are true, and R is the correct explanation of A.
Rhombus (all sides equal) + all angles 90° = Square.
(Logical Reasoning)
20. Assertion (A): Construction lines are drawn lightly in geometric constructions.
Reason (R): They are temporary guides to be erased later.
Reason (R): They are temporary guides to be erased later.
Answer: a) Both A and R are true, and R is the correct explanation of A.
Light lines are used because they are temporary, as explained in R.
(Procedural Understanding)
C. True/False (10 Questions)
1. All rectangles are squares.
Answer: False
All squares are rectangles, but not all rectangles are squares.
(Conceptual Clarity)
2. A square has 4 lines of symmetry.
Answer: True
Two diagonals and two lines through midpoints of opposite sides.
(Spatial Understanding)
3. The diagonals of a square are equal.
Answer: True
This is a key property of squares and rectangles.
(Factual Recall)
4. A quadrilateral with all angles 90° must be a square.
Answer: False
Could be a rectangle if sides are not all equal.
(Conceptual Understanding)
5. A rectangle can be constructed if only one side is known.
Answer: False
Need both adjacent sides or one side and diagonal.
(Procedural Understanding)
6. In a rectangle, opposite sides are parallel.
Answer: True
This is a fundamental property of rectangles.
(Factual Recall)
7. A rhombus is a square if one angle is 90°.
Answer: True
If one angle is 90°, all become 90° in a rhombus.
(Logical Reasoning)
8. A rotated rectangle is still a rectangle.
Answer: True
Rotation preserves lengths and angles.
(Spatial Understanding)
9. A compass can only draw full circles.
Answer: False
Can draw arcs (parts of circles).
(Tool Knowledge)
10. In a square, diagonals are perpendicular to each other.
Answer: True
Diagonals of a square intersect at right angles.
(Factual Recall)
D. Short Answer Type I (2 Marks each – 15 Questions)
1. Write two properties of a rectangle.
Answer:
1. Opposite sides are equal in length.
2. All interior angles are right angles (90°).
2. All interior angles are right angles (90°).
(Recall, Basic Application)
2. If a rectangle has length 8 cm and breadth 6 cm, what is the length of its diagonal?
Answer: 10 cm
Using Pythagoras theorem: √(8² + 6²) = √(64 + 36) = √100 = 10 cm.
(Problem-Solving, Numerical Application)
3. How many different ways can you name a rectangle with vertices W, X, Y, Z?
Answer: 8 ways
Start from any of 4 vertices and go clockwise or anti-clockwise: 4 × 2 = 8.
(Analytical Thinking)
4. How do you draw a perpendicular to a line using a compass?
Answer:
1. Mark point P on line. 2. With P as center, draw arcs cutting line at A and B. 3. With A and B as centers, draw intersecting arcs above/below line at Q. 4. Join P and Q.
(Procedural Application)
5. Write one similarity and one difference between a square and a rectangle.
Answer:
Similarity: Both have all angles 90°.
Difference: Square has all sides equal; rectangle has only opposite sides equal.
Difference: Square has all sides equal; rectangle has only opposite sides equal.
(Comparative Analysis)
6. In a rectangle ABCD, if AB = 7 cm and BC = 5 cm, what are the lengths of CD and AD?
Answer: CD = 7 cm, AD = 5 cm
Opposite sides are equal: CD opposite AB, AD opposite BC.
(Logical Reasoning)
7. What is the minimum number of measurements needed to construct a square?
Answer: 1
Only side length is needed.
(Conceptual Understanding)
8. If a diagonal of a square is 10 cm, what is the side length?
Answer: 10/√2 = 5√2 cm ≈ 7.07 cm
Diagonal = side × √2, so side = diagonal/√2.
(Numerical Application)
9. Can a rectangle be divided into two identical squares? If yes, give an example of side lengths.
Answer: Yes. Example: 4 cm × 2 cm rectangle.
Length must be exactly twice the breadth.
(Analytical Thinking)
10. What is the purpose of drawing light construction lines?
Answer:
To guide accurate construction; they are temporary and erased later.
(Procedural Understanding)
11. What is the first step in constructing a square of side 6 cm?
Answer: Draw a line segment of 6 cm.
This forms the base side of the square.
(Procedural Application)
12. How do you verify that opposite sides of a constructed rectangle are equal?
Answer:
Measure the lengths using a ruler and compare opposite sides.
(Verification Skills)
13. What happens to a square when it is rotated?
Answer: It remains a square.
Rotation doesn't change side lengths or angles.
(Spatial Understanding)
14. In the "Wavy Wave", if central line is 10 cm, what is length AX for half-circle wave?
Answer: 5 cm
X is midpoint, so AX = half of AB = 5 cm.
(Application)
15. Name the tool used to draw arcs in geometric constructions.
Answer: Compass
Compass is specifically designed for drawing arcs and circles.
(Tool Knowledge)
E. Short Answer Type II (3 Marks each – 10 Questions)
1. Construct a rectangle with sides 5 cm and 3 cm. Verify its properties.
Answer:
Steps:
1. Draw AB = 5 cm.
2. Construct perpendiculars at A and B.
3. Mark AD = 3 cm on perpendicular from A.
4. Mark BC = 3 cm on perpendicular from B.
5. Join D and C.
Verification:
• Measure opposite sides: AB = CD = 5 cm, BC = AD = 3 cm. ✔
• Measure angles: ∠A = ∠B = ∠C = ∠D = 90°. ✔
1. Draw AB = 5 cm.
2. Construct perpendiculars at A and B.
3. Mark AD = 3 cm on perpendicular from A.
4. Mark BC = 3 cm on perpendicular from B.
5. Join D and C.
Verification:
• Measure opposite sides: AB = CD = 5 cm, BC = AD = 3 cm. ✔
• Measure angles: ∠A = ∠B = ∠C = ∠D = 90°. ✔
(Problem-Solving, Procedural Application, Verification)
2. Explain how to locate a point that is 4 cm from point P and 4 cm from point Q using compass.
Answer:
1. Open compass to 4 cm radius.
2. With P as center, draw an arc.
3. With Q as center, draw another arc of same radius.
4. The intersection(s) of the arcs are the required point(s) equidistant from P and Q.
2. With P as center, draw an arc.
3. With Q as center, draw another arc of same radius.
4. The intersection(s) of the arcs are the required point(s) equidistant from P and Q.
(Procedural Application, Spatial Understanding)
3. Draw a square of side 4 cm without using a protractor.
Answer:
Steps (using compass and ruler):
1. Draw PQ = 4 cm.
2. At P, construct a perpendicular line.
3. With P as center, radius 4 cm, draw arc on perpendicular to mark S (PS=4cm).
4. At Q, construct a perpendicular line.
5. With Q as center, radius 4 cm, draw arc on this perpendicular to mark R (QR=4cm).
6. Join R and S.
1. Draw PQ = 4 cm.
2. At P, construct a perpendicular line.
3. With P as center, radius 4 cm, draw arc on perpendicular to mark S (PS=4cm).
4. At Q, construct a perpendicular line.
5. With Q as center, radius 4 cm, draw arc on this perpendicular to mark R (QR=4cm).
6. Join R and S.
(Procedural Application, Tool Usage)
4. Divide a rectangle of sides 9 cm and 3 cm into three identical squares.
Answer:
Steps:
1. Construct rectangle ABCD with AB = 9 cm, BC = 3 cm.
2. On AB, mark points E and F such that AE = EF = FB = 3 cm each.
3. Draw lines from E and F perpendicular to AB (parallel to BC) to meet CD.
4. This divides rectangle into three 3 cm × 3 cm squares.
1. Construct rectangle ABCD with AB = 9 cm, BC = 3 cm.
2. On AB, mark points E and F such that AE = EF = FB = 3 cm each.
3. Draw lines from E and F perpendicular to AB (parallel to BC) to meet CD.
4. This divides rectangle into three 3 cm × 3 cm squares.
(Problem-Solving, Application)
5. In a rectangle, one diagonal divides an angle into 55° and 35°. What are the other angles?
Answer: 90°, 55°, and 35°
• The full angle at that vertex is 90° (55°+35°).
• The opposite vertex also has its angle divided into 55° and 35°.
• The remaining two vertices each have 90° angles.
• The opposite vertex also has its angle divided into 55° and 35°.
• The remaining two vertices each have 90° angles.
(Logical Reasoning, Analytical Thinking)
6. Construct a rectangle with one side 6 cm and diagonal 10 cm.
Answer:
Steps:
1. Draw base CD = 6 cm.
2. At C, draw perpendicular line `l`.
3. With D as center, radius 10 cm, draw arc cutting line `l` at B.
4. With B as center, radius 6 cm, draw arc.
5. With D as center, radius = BC (measure BC first), draw arc to intersect previous arc at A.
6. Join A to B and A to D.
1. Draw base CD = 6 cm.
2. At C, draw perpendicular line `l`.
3. With D as center, radius 10 cm, draw arc cutting line `l` at B.
4. With B as center, radius 6 cm, draw arc.
5. With D as center, radius = BC (measure BC first), draw arc to intersect previous arc at A.
6. Join A to B and A to D.
(Procedural Application, Problem-Solving)
7. How will you draw the "Wavy Wave" pattern using a compass?
Answer:
Steps:
1. Draw central line AB of desired length (e.g., 8 cm).
2. Find midpoint X of AB.
3. For half-circle wave above line: With A as center, radius AX, draw semicircle from A to X.
4. For wave below line: With X as center, same radius, draw semicircle from X to B.
For smaller waves: Adjust compass radius to less than AX.
1. Draw central line AB of desired length (e.g., 8 cm).
2. Find midpoint X of AB.
3. For half-circle wave above line: With A as center, radius AX, draw semicircle from A to X.
4. For wave below line: With X as center, same radius, draw semicircle from X to B.
For smaller waves: Adjust compass radius to less than AX.
(Procedural Application, Creativity)
8. Using a compass, bisect a line segment of length 8 cm.
Answer:
Steps:
1. Draw PQ = 8 cm.
2. With P as center, radius > 4 cm, draw arcs above and below PQ.
3. With Q as center, same radius, draw arcs to intersect first arcs at R (above) and S (below).
4. Draw line RS. Intersection M with PQ is the midpoint, bisecting PQ into 4 cm each.
1. Draw PQ = 8 cm.
2. With P as center, radius > 4 cm, draw arcs above and below PQ.
3. With Q as center, same radius, draw arcs to intersect first arcs at R (above) and S (below).
4. Draw line RS. Intersection M with PQ is the midpoint, bisecting PQ into 4 cm each.
(Procedural Application, Tool Usage)
9. Construct a rectangle that can be divided into two identical squares.
Answer:
Steps:
1. Choose square side, e.g., 3 cm. Rectangle will be 3 cm × 6 cm.
2. Draw AB = 6 cm.
3. Construct perpendiculars at A and B.
4. Mark AD = BC = 3 cm.
5. Join D and C.
6. Find midpoint M of AB. Draw line through M parallel to AD to divide into two squares.
1. Choose square side, e.g., 3 cm. Rectangle will be 3 cm × 6 cm.
2. Draw AB = 6 cm.
3. Construct perpendiculars at A and B.
4. Mark AD = BC = 3 cm.
5. Join D and C.
6. Find midpoint M of AB. Draw line through M parallel to AD to divide into two squares.
(Problem-Solving, Application)
10. Show steps to construct a square when its diagonal is given as 8 cm.
Answer:
Steps:
1. Draw diagonal AC = 8 cm.
2. Find midpoint O of AC.
3. At O, construct perpendicular bisector of AC.
4. On perpendicular bisector, mark points B and D such that OB = OD = 4 cm (half of diagonal).
5. Join A, B, C, D to form square.
(Note: side = diagonal/√2 = 8/√2 = 4√2 ≈ 5.66 cm)
1. Draw diagonal AC = 8 cm.
2. Find midpoint O of AC.
3. At O, construct perpendicular bisector of AC.
4. On perpendicular bisector, mark points B and D such that OB = OD = 4 cm (half of diagonal).
5. Join A, B, C, D to form square.
(Note: side = diagonal/√2 = 8/√2 = 4√2 ≈ 5.66 cm)
(Problem-Solving, Procedural Application)
F. Long Answer Type (5 Marks each – 10 Questions)
1. Construct a square of side 6 cm. Measure its diagonals and verify they are equal.
Answer:
Construction Steps:
1. Draw line segment PQ = 6 cm.
2. At P, construct perpendicular line. Mark point S on it such that PS = 6 cm.
3. At Q, construct perpendicular line. Mark point R on it such that QR = 6 cm.
4. Join R and S. PQRS is the constructed square.
Verification:
• Using a ruler, measure diagonals PR and QS.
• Observation: Both diagonals measure approximately 8.5 cm (6 × √2 ≈ 8.49 cm).
• Conclusion: The diagonals are equal in length.
1. Draw line segment PQ = 6 cm.
2. At P, construct perpendicular line. Mark point S on it such that PS = 6 cm.
3. At Q, construct perpendicular line. Mark point R on it such that QR = 6 cm.
4. Join R and S. PQRS is the constructed square.
Verification:
• Using a ruler, measure diagonals PR and QS.
• Observation: Both diagonals measure approximately 8.5 cm (6 × √2 ≈ 8.49 cm).
• Conclusion: The diagonals are equal in length.
(Comprehensive Problem-Solving, Measurement, Verification)
2. Construct a rectangle ABCD with AB = 8 cm and BC = 5 cm. Draw its diagonals and measure the angles they make with the sides.
Answer:
Construction Steps:
1. Draw AB = 8 cm.
2. Construct perpendiculars at A and B.
3. Mark AD = 5 cm on perpendicular from A.
4. Mark BC = 5 cm on perpendicular from B.
5. Join D and C.
6. Draw diagonals AC and BD.
Measurement & Observation:
• Using protractor, measure angles like ∠CAB, ∠CAD, ∠ABD, etc.
• Diagonals are not perpendicular to sides (unlike square).
• Specific angles can be calculated: tan θ = opposite/adjacent side ratio.
1. Draw AB = 8 cm.
2. Construct perpendiculars at A and B.
3. Mark AD = 5 cm on perpendicular from A.
4. Mark BC = 5 cm on perpendicular from B.
5. Join D and C.
6. Draw diagonals AC and BD.
Measurement & Observation:
• Using protractor, measure angles like ∠CAB, ∠CAD, ∠ABD, etc.
• Diagonals are not perpendicular to sides (unlike square).
• Specific angles can be calculated: tan θ = opposite/adjacent side ratio.
(Comprehensive Construction, Measurement, Analysis)
3. Construct the "House" figure with all sides 5 cm. Show all construction arcs.
Answer:
Detailed Steps:
1. Base: Draw horizontal line BC = 5 cm.
2. Locate A (Roof Peak):
• With B as center, radius 5 cm, draw arc above BC.
• With C as center, radius 5 cm, draw another arc to intersect first arc at A.
3. Walls: Join A to B and A to C. (AB = AC = 5 cm).
4. Door: Mark point in middle of BC. Draw rectangular door (e.g., 2 cm wide).
5. Roof Arc: With A as center, radius 5 cm, draw circular arc from B to C.
Key: Show construction arcs lightly to demonstrate how points were located.
1. Base: Draw horizontal line BC = 5 cm.
2. Locate A (Roof Peak):
• With B as center, radius 5 cm, draw arc above BC.
• With C as center, radius 5 cm, draw another arc to intersect first arc at A.
3. Walls: Join A to B and A to C. (AB = AC = 5 cm).
4. Door: Mark point in middle of BC. Draw rectangular door (e.g., 2 cm wide).
5. Roof Arc: With A as center, radius 5 cm, draw circular arc from B to C.
Key: Show construction arcs lightly to demonstrate how points were located.
(Creative Construction, Procedural Application, Spatial Design)
4. Construct a rectangle that can be divided into two identical squares. Explain your steps.
Answer:
Explanation: For rectangle to be divisible into two squares, length must be twice breadth.
Construction Steps:
1. Plan: Choose square side, e.g., 3 cm. Rectangle: 3 cm × 6 cm.
2. Construct Rectangle:
• Draw AB = 6 cm (length).
• Construct perpendiculars at A and B.
• Mark AD = BC = 3 cm (breadth).
• Join DC. Rectangle ABCD ready.
3. Show Division:
• Find midpoint M of AB.
• Draw line through M parallel to AD (or perpendicular to AB).
• This line divides rectangle into two 3 cm × 3 cm squares.
Construction Steps:
1. Plan: Choose square side, e.g., 3 cm. Rectangle: 3 cm × 6 cm.
2. Construct Rectangle:
• Draw AB = 6 cm (length).
• Construct perpendiculars at A and B.
• Mark AD = BC = 3 cm (breadth).
• Join DC. Rectangle ABCD ready.
3. Show Division:
• Find midpoint M of AB.
• Draw line through M parallel to AD (or perpendicular to AB).
• This line divides rectangle into two 3 cm × 3 cm squares.
(Planning, Construction, Analysis)
5. Construct a rectangle with one side 7 cm and a diagonal 9 cm. Verify rectangle properties.
Answer:
Construction Steps:
1. Draw side CD = 7 cm.
2. At C, draw perpendicular line `l`.
3. With D as center, radius 9 cm, draw arc cutting line `l` at B.
4. Now we have C, D, B. To find A:
• Method 1: Draw line through D parallel to CB.
• Method 2: With B as center, radius 7 cm, and with D as center, radius = BC, draw intersecting arcs at A.
5. Join A to B and A to D.
Verification:
• Measure opposite sides: CD = AB? AD = BC? (Should be equal).
• Measure angles: Check if ∠C, ∠D, ∠A, ∠B are 90° using protractor.
1. Draw side CD = 7 cm.
2. At C, draw perpendicular line `l`.
3. With D as center, radius 9 cm, draw arc cutting line `l` at B.
4. Now we have C, D, B. To find A:
• Method 1: Draw line through D parallel to CB.
• Method 2: With B as center, radius 7 cm, and with D as center, radius = BC, draw intersecting arcs at A.
5. Join A to B and A to D.
Verification:
• Measure opposite sides: CD = AB? AD = BC? (Should be equal).
• Measure angles: Check if ∠C, ∠D, ∠A, ∠B are 90° using protractor.
(Comprehensive Construction, Verification, Problem-Solving)
6. Construct a square with 8 cm side. Draw lines joining midpoints of opposite sides.
Answer:
Construction Steps:
1. Construct square PQRS with side 8 cm.
2. Find midpoints of each side:
• M = midpoint of PQ
• N = midpoint of QR
• O = midpoint of RS
• P = midpoint of SP
3. Draw lines:
• Join M to O (through center, horizontal)
• Join N to P (through center, vertical)
Observation:
• These lines intersect at center of square.
• They divide square into 4 equal smaller squares.
• They are perpendicular to each other.
1. Construct square PQRS with side 8 cm.
2. Find midpoints of each side:
• M = midpoint of PQ
• N = midpoint of QR
• O = midpoint of RS
• P = midpoint of SP
3. Draw lines:
• Join M to O (through center, horizontal)
• Join N to P (through center, vertical)
Observation:
• These lines intersect at center of square.
• They divide square into 4 equal smaller squares.
• They are perpendicular to each other.
(Construction, Analysis, Spatial Understanding)
7. Construct a rectangle where one diagonal divides opposite angles into 60° and 30°.
Answer:
Planning: At two opposite vertices, the 90° angle is split by diagonal into 60° and 30°.
Construction Steps:
1. Draw base AB of convenient length.
2. At A, construct 60° angle (using compass or protractor).
3. At A, also construct perpendicular to AB (this will be side AD).
4. The diagonal lies between AB and AD, making 60° with AB and 30° with AD.
5. Choose point D on perpendicular from A.
6. From D, draw line parallel to AB. From B, draw line parallel to AD.
7. Their intersection is C, completing rectangle.
Verification: Measure angles at A and C to ensure diagonal divides them into 60° and 30°.
Construction Steps:
1. Draw base AB of convenient length.
2. At A, construct 60° angle (using compass or protractor).
3. At A, also construct perpendicular to AB (this will be side AD).
4. The diagonal lies between AB and AD, making 60° with AB and 30° with AD.
5. Choose point D on perpendicular from A.
6. From D, draw line parallel to AB. From B, draw line parallel to AD.
7. Their intersection is C, completing rectangle.
Verification: Measure angles at A and C to ensure diagonal divides them into 60° and 30°.
(Advanced Construction, Planning, Verification)
8. Construct a "Wavy Wave" with central line 10 cm and half-circle waves.
Answer:
Construction Steps:
1. Draw central line AB = 10 cm.
2. Find midpoint X of AB (AX = XB = 5 cm).
3. First Wave (Half-circle above line):
• Place compass tip on A.
• Set radius to AX = 5 cm.
• Draw semicircular arc from A to X, above line AB.
4. Second Wave (Half-circle below line):
• Place compass tip on X.
• Keep same radius (5 cm).
• Draw semicircular arc from X to B, below line AB.
Result: Continuous wave pattern with alternating half-circles.
1. Draw central line AB = 10 cm.
2. Find midpoint X of AB (AX = XB = 5 cm).
3. First Wave (Half-circle above line):
• Place compass tip on A.
• Set radius to AX = 5 cm.
• Draw semicircular arc from A to X, above line AB.
4. Second Wave (Half-circle below line):
• Place compass tip on X.
• Keep same radius (5 cm).
• Draw semicircular arc from X to B, below line AB.
Result: Continuous wave pattern with alternating half-circles.
(Creative Construction, Procedural Application)
9. Draw two identical "Eyes" using compass construction.
Answer:
Strategy: Each eye = two symmetrical curves (upper and lower eyelid).
Construction Steps for One Eye:
1. Draw light horizontal guide line.
2. Mark two points L and R about 3-4 cm apart as ends of eye.
3. Find center points U (for upper curve) and D (for lower curve) vertically aligned.
4. U should be above guide line, D below it, on same vertical line.
5. With U as center, draw arc from L to R for upper curve.
6. With D as center, draw arc from L to R for lower curve.
For Second Eye:
• Repeat same process next to first eye.
• Ensure same size, alignment, and symmetry.
Key: Practice needed for good symmetry.
Construction Steps for One Eye:
1. Draw light horizontal guide line.
2. Mark two points L and R about 3-4 cm apart as ends of eye.
3. Find center points U (for upper curve) and D (for lower curve) vertically aligned.
4. U should be above guide line, D below it, on same vertical line.
5. With U as center, draw arc from L to R for upper curve.
6. With D as center, draw arc from L to R for lower curve.
For Second Eye:
• Repeat same process next to first eye.
• Ensure same size, alignment, and symmetry.
Key: Practice needed for good symmetry.
(Creative Construction, Symmetry, Precision)
10. Construct a square and then divide it into four equal smaller squares.
Answer:
Construction Steps:
1. Construct square PQRS of desired side, e.g., 8 cm.
2. Find midpoints of each side:
• M = midpoint of PQ
• N = midpoint of QR
• O = midpoint of RS
• T = midpoint of SP
3. Draw lines:
• Join M to O (horizontal through center)
• Join N to T (vertical through center)
Result:
• Square divided into 4 equal smaller squares.
• Each small square has side = half of original = 4 cm.
• All lines intersect at center of original square.
Verification: Measure sides of small squares to confirm equality.
1. Construct square PQRS of desired side, e.g., 8 cm.
2. Find midpoints of each side:
• M = midpoint of PQ
• N = midpoint of QR
• O = midpoint of RS
• T = midpoint of SP
3. Draw lines:
• Join M to O (horizontal through center)
• Join N to T (vertical through center)
Result:
• Square divided into 4 equal smaller squares.
• Each small square has side = half of original = 4 cm.
• All lines intersect at center of original square.
Verification: Measure sides of small squares to confirm equality.
(Construction, Division, Verification)
G. Case-Based Questions (5 Cases, each with 4 Sub-Questions)
Case 1: A rectangle has vertices A, B, C, D. AB = 6 cm, BC = 4 cm. Diagonals AC and BD intersect at O.
1. What is the length of CD?
Answer: b) 6 cm
CD is opposite to AB in rectangle ABCD.
(Data Interpretation, Applied Problem-Solving)
2. What is the length of diagonal AC?
Answer: a) 7.2 cm (approximately)
AC² = AB² + BC² = 6² + 4² = 36 + 16 = 52, AC = √52 ≈ 7.21 cm.
(Analytical Thinking, Numerical Application)
3. If ∠CAB = 30°, then ∠ACB =
Answer: b) 60°
In ΔABC, ∠B = 90°, ∠CAB = 30°, so ∠ACB = 180° - (90° + 30°) = 60°.
(Geometric Reasoning)
4. How many pairs of equal triangles are formed by the diagonals?
Answer: b) 4
Four triangles: ΔAOB, ΔBOC, ΔCOD, ΔDOA. They are equal in two pairs: ΔAOB ≅ ΔCOD and ΔBOC ≅ ΔDOA.
(Analytical Thinking)
Case 2: A square sheet of side 10 cm is rotated to look like a diamond.
1. Is it still a square?
Answer: a) Yes
Rotation doesn't change side lengths or angles.
(Spatial Understanding)
2. What is the length of each side after rotation?
Answer: b) Remains 10 cm
Rotation is rigid transformation preserving lengths.
(Conceptual Understanding)
3. What is the angle between two adjacent sides after rotation?
Answer: b) 90°
Internal angles don't change with rotation.
(Spatial Understanding)
4. How many lines of symmetry does it have now?
Answer: c) 4
Square always has 4 lines of symmetry regardless of orientation.
(Spatial Understanding)
Case 3: In the "House" construction, all edges are 5 cm.
1. How many arcs are needed to locate point A?
Answer: b) 2
One arc from B (radius 5 cm) and one from C (radius 5 cm).
(Procedural Understanding)
2. What is the shape of the roof?
Answer: c) Circular arc
Roof is drawn as arc of circle with center A, radius 5 cm.
(Spatial Understanding)
3. Which tool is essential for this construction?
Answer: b) Compass
Compass needed for drawing arcs of radius 5 cm.
(Tool Knowledge)
4. The base BC is of length:
Answer: a) 5 cm
All edges are 5 cm, including base BC.
(Data Interpretation)
Case 4: A rectangle is divided into 3 identical squares.
1. If the shorter side of rectangle is 4 cm, the longer side is:
Answer: b) 12 cm
Longer side = 3 × shorter side = 3 × 4 = 12 cm.
(Analytical Thinking)
2. How many squares are formed in total?
Answer: b) 3
Case states rectangle is divided into 3 identical squares.
(Data Interpretation)
3. What is the perimeter of each small square?
Answer: c) 16 cm
Each square side = shorter side of rectangle = 4 cm. Perimeter = 4 × 4 = 16 cm.
(Numerical Application)
4. Can this rectangle be a square?
Answer: b) No
For square, length = breadth. Here 12 cm ≠ 4 cm.
(Logical Reasoning)
Case 5: Constructing rectangle given side and diagonal.
1. If side = 5 cm and diagonal = 13 cm, what is the other side?
Answer: c) 12 cm
Other side = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.
(Numerical Application)
2. Which tool is used to draw the arc with diagonal as radius?
Answer: c) Compass
Compass draws arcs of given radius.
(Tool Knowledge)
3. How many perpendicular lines are drawn in this construction?
Answer: b) 2
One at end of given side, and possibly one more to complete rectangle.
(Procedural Understanding)
4. What property of rectangle is used to verify the construction?
Answer: d) All of these
All rectangle properties should be verified after construction.
(Verification Skills)
