Two line segments AB and AC include an angle of 60° where AB = 5 cm and AC = 7 cm. 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.

#### Solution

**Steps of construction:**

**1.** Draw a line segment AB = 5 cm.**2.** Draw ∠BAZ = 60°.

**3.** With centre A and radius 7 cm, draw an arc cutting the line AZ at C.

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

**5.** Divide AX into four equal parts, namely AA_{1 }= A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4}.

**6.** Join A_{4}B.

**7.** Draw A_{3}P || A_{4}B meeting AB at P.

**8.** Hence, we obtain, P is the point on AB such that AP = `3/4` AB.

**9.** Next, draw a ray AY, such that it makes an acute ∠CAY.

**10.** Divide AY into four parts, namely AB_{1} = B_{1}B_{2 }= B_{2}B_{3} = B_{3}B_{4}.

**11.** Join B_{4}C.

**12.** Draw B_{1}Q || B_{4}C meeting AC at Q. We get, Q is the point on AC such that AQ = `1/4` AC.

**13.** Join PQ and measure it.

**14.** PQ = 3.25 cm