Two-line segment AB and AC include an angle of 60°, where AB=5cm and AC=7cm. Locate points P and Q on AB and AC, respectively such that AP= 3/4 AB and AQ= 1/4 AC. Join P and Q and measure the length PQ.

Given,

AB=5cm

AC=7cm

AP = 3/4 AB and AQ = 1/2 AC … (i)

From equation-(i)

AP = 3/4 × AB = 3/4 × 5 = 15/4 cm

P is any point on the AB

PB=AB-AP

PB = 5 – 15/4

= 5/4 cm

∴ AP: PB = 15/4 : 5/4

⇒ AP: PB = 3: 1

i.e. the scale factor of line segment AB is 3/1.

From Eq. (i).

AQ=1/4 AC

= 1/4 × 7 = 7/4 cm

Q is any point on the AC

QC = AC – AQ

QC = 7 – 7/4

_{= 21/4 cm}

∴ AQ : QC = 7/4 : 21/4 = 1:3

⇒ AQ : QC = 1: 3

i.e. scale factor of line segment AQ is 1/3.

Steps of construction

1. Draw a line AB=5cm.

2. Draw a ray AZ making an acute angle, ∠ BAZ=60°.

3. With A as center and radius equal to 7 cm draw an arc cutting the line AZ at C.

4. Draw a ray AX, (make acute ∠BAX).

5. Along AX, mark 4 points A₁, A₂, A₃, and A₄

Such that A₁A₂₌A₁A₃₌A₃A₄

6. Join A₄B

7. Draw A₃P||A₄B meeting AB at P.

[by making an angle equal to ∠AA₄ B]

Then, P is the point on AB which divides it in the ratio 3:1.

So, AP: PB=3:1

8. Draw a ray AY, making an acute ∠CAY.

9. Along AY, mark 4 point B₁B₂, B₃and B_4.

Such that AB₁=B₁B₂=B₂B₃=B₃B₄

10. Join B₄C.

11. Draw B₁Q||B₄C meeting AC at Q.

[by making an angle equal to∠AB₄C]

Then, Q is the point on AC which divides it in the ratio1:3.

So, AQ:QC=1:3

12. Finally, join PQ and its measurement is 3.25cm.