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.

Thinking process:

I. Firstly, we find the ratio of AB in which P divides it with the help of the relation A= 3/4 AB.

II. Secondly, we find the ratio of AC in which Q divides it with the help of the relation AQ= 1/4 AC.

III. Now, construct the line segment AB and AC in which P and Q respectively divides it in the ratio from step
(i) and (ii) respectively.

IV. Finally get the points P and Q. After that join PQ and get the required measurement of PQ.

Given that, AB=5cm and AC=7cm

Also, and … (i)

From Eq.(i)

Then, PB=AB-AP

[As P is any point on the AB]

Hence AP: PB = 3:1

i.e. , scale factor of line segment AB is .

Again from Eq. (i).

Then,

[As Q is any point on the AC]

Hence AQ:QC = 1:3

i.e. scale factor of line segment AQ is .

Steps of construction:

1. Draw a line segment AB=5cm.

2. Now draw a ray AZ making an acute ∠BAZ=60°.

a. With A as centre and with any radius, draw another arc cutting the line AB at D.

b. With D as centre and with the same radius, draw an arc cutting the first arc (drawn in step a) at point E.

c. Now join the ray AZ which forms an angle of 60° with the line AB.

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

4. Draw a ray AX, making an acute ∠BAX.

5. Along AX, mark 1+3=4 points A₁,A₂,A₃, and A₄

Such that A₁A₂₌A₁A₃₌A₃A₄

6. Join A₄B

7. From A₃ 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 3+1=4 point B₁B₂,B₃ and B_4.

Such that AB₁=B₁B₂=B₂B₃=B₃B₄

10. Join B₄C.

11. From B₁ 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.