Question 1

Task: Draw line segment AB = 10 cm. Construct its perpendicular bisector and mark the midpoint M. State AM and MB.

Solution (steps):

1. Use a ruler to draw AB = 10 cm.


2. Put the compass point at A. Open it to a radius slightly more than 5 cm (say 6 cm). Draw an arc above and below AB.


3. Without changing the compass width, place the compass at B and draw arcs above and below so they cut the first arcs. Call the upper intersection X and lower intersection Y.


4. Use the ruler to join X and Y. The line XY meets AB at M.


5. Measure AM and MB with the ruler: each is 5 cm.

Diagram:

   X         Y
   *         *
A-----M-----B
  

Answer: AM = MB = 5 cm.

Question 2

Task: Draw line l. From a point P not on l, construct a line through P that is parallel to l. Show how to check the lines are parallel.

Solution (steps):

1. Draw line l. Mark point P above the line.


2. From P, draw a perpendicular to l (use the perpendicular-from-point method). Call this line PR.


3. At P, construct a perpendicular to PR. That second perpendicular is parallel to l. Name it m.


4. To check: extend l and m; if they never meet and the distance between them is the same at two places, they are parallel. Or check both are perpendicular to the same line PR.

Diagram:

P----------m
 (perpendicular to PR then back)
l: A--------B
  

Answer: Line through P is parallel to l (m ∥ l).

Question 3

Task: Construct triangle XYZ with XY = 6 cm, angle XYZ = 60°, and YZ = 5 cm.

Solution (steps):

1. Draw base XY = 6 cm.


2. At point Y, construct an angle of 60° (use compass method). Draw the ray for this angle.


3. On that ray from Y, measure and mark point Z so that YZ = 5 cm (use compass to transfer 5 cm).


4. Join X to Z. Triangle XYZ is complete.

Diagram:

   Z
  /
 /   (YZ = 5 cm)
Y-----X (XY = 6 cm)
  

Answer: Triangle XYZ constructed; check XY = 6 cm, YZ = 5 cm, ∠XYZ = 60°.

Question 4

Task: Construct a square of side 6 cm. Measure its diagonal AC.

Solution (steps):

1. Draw AB = 6 cm.


2. At A draw a perpendicular. On that perpendicular mark AD = 6 cm.


3. At B draw a perpendicular (same direction) and on it mark BC = 6 cm.


4. Join CD to complete square ABCD.


5. Measure diagonal AC with ruler or by calculation: AC = √(6² + 6²) ≈ √72 ≈ 8.49 cm (≈ 8.5 cm).

Diagram:

D-----C
|     |
|     |
A-----B
(side = 6 cm)
  

Answer: Diagonal AC ≈ 8.5 cm (rounded).

Question 5

Task: Construct an equilateral triangle of side 7 cm. Measure one angle to confirm.

Solution (steps):

1. Draw AB = 7 cm.


2. With compass at A radius 7 cm, draw an arc. With compass at B radius 7 cm, draw another arc meeting the first at C.


3. Join AC and BC. Triangle ABC is equilateral.


4. Measure angle at A with a protractor (or know from geometry): each angle = 60°.

Diagram:

   C
  / \
 /   \
A-----B
(side = 7 cm)
  

Answer: Each angle = 60°.

Question 6

Task: On a line draw point O. Construct two lines through O that are perpendicular to each other (i.e., make a cross). Then label the four right angles.

Solution (steps):

1. Draw a straight line and mark point O on it.


2. Using the perpendicular-at-point method at O, construct a line PQ through O that is perpendicular to the first line.


3. The two lines cross at O and form four right angles. Label them ∠1, ∠2, ∠3, ∠4 around O.

Diagram:

   |
   |
---O---
   |
   |
  

Answer: Four right angles at O; each = 90°.

Question 7

Task: Draw segment KL = 7 cm. Bisect KL and name midpoint M. Then show KM = ML.

Solution (steps):

1. Draw KL = 7 cm.


2. With compass at K radius > 3.5 cm, draw arcs above and below. Without changing width, draw same arcs from L to intersect the first arcs at X and Y.


3. Join X and Y; where XY meets KL is M.


4. Measure KM and ML: both = 3.5 cm.

Diagram:

*       *
K---M---L
*       *
  

Answer: KM = ML = 3.5 cm.

Question 8

Task: Construct angle 90° at point A on line AB. Then bisect that right angle to form 45° angles. Show steps.

Solution (steps):

1. Draw AB and mark A. Use perpendicular-at-point method to construct AD so that AD ⟂ AB (so ∠DAB = 90°).


2. To bisect ∠DAB: with centre A draw an arc that cuts AD and AB at points P and Q.


3. With same radius draw arcs from P and Q so they meet at R.


4. Join A to R. AR bisects the 90° angle.


5. Each part is 45°.

Diagram:

    \
     \  (45°)
A----+----
     /  (45°)
    /
  

Answer: Two angles each = 45°.

Question 9

Task: Given triangle ABC, construct the perpendicular bisector of side AB and note where it meets AB. Explain why that point is the midpoint.

Solution (steps):

1. In triangle ABC, draw arcs from A and B with equal radius (greater than half AB) to meet at X and Y above and below AB.


2. Join X and Y to form the perpendicular bisector. Where this bisector crosses AB is M.


3. Because arcs were equal from A and B, the bisector is equidistant from A and B; therefore it cuts AB into two equal parts, so AM = MB.

Diagram:

   X     Y
   *     *
A--M--B
   *     *
  

Answer: M is midpoint because it is equidistant from A and B.

Question 10

Task: Construct triangle PQR given PQ = 5 cm, PR = 4 cm, and angle QPR = 60°. Show construction and final triangle.

Solution (steps):

1. Draw ray PR. From point P on that ray, measure PR = 4 cm and mark R. (Alternatively draw PR = 4 cm first.)


2. At P, construct angle QPR = 60° (use compass). Draw the 60° ray.


3. On that 60° ray from P, measure PQ = 5 cm and mark Q.


4. Join Q to R. Triangle PQR is complete with given sides and angle.

Diagram:

   Q
  /
 /   (PQ = 5 cm)
P-----R (PR = 4 cm)
  

Answer: Triangle constructed; check PQ = 5 cm, PR = 4 cm, ∠QPR = 60°.

Teaching notes for students (simple reminders)

• Always draw light construction arcs first. Darken the final lines only after checking.


• Keep the compass width steady when the steps say “without changing the compass width.”


• Label every intersection point clearly — those are the places you join with the ruler.


• For angle checks you may use a protractor while learning, but practice to rely on compass methods.


• Practice slowly; neatness matters in construction.