##### 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 B4.

Such that AB1=B1B2=B2B3=B3B4

10. Join B4C.

11. Draw B1Q||B4C meeting AC at Q.

[by making an angle equal toAB4C]

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.

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