Question

In ΔPQR, ∠P : ∠Q : ∠R = 1 : 2 : 3 and altitude from P to QR is 4 cm. Find the perimeter of ΔPQR.

Answer 1

Given:

Angles are in ratio ∠P : ∠Q : ∠R = 1 : 2 : 3

Altitude from P to base QR = 4 cm

Find the perimeter of △PQR

Step 1: 

Let ∠P = x, ∠Q = 2x, ∠R = 3x

x + 2x + 3x = 6x = 180°

x = 30°

∠P = 30°

∠Q = 60°

∠R = 90°

Step 2:

Since ∠R = 90°, the triangle is right-angled at R.

Let, Height from P to base QR = 4 cm

Use triangle △PQR with ∠P = 30°

sin(30°) = `"opposite"/"hypotenuse"`

= `4/(PQ) = 4/sin(30°)`

= `4/(1/2)` = 8 cm

cos(30°) = `"adjacent (QR)"/"hypotenuse (PQ)"`

cos(30°) = `(QR)/8`

= QR = 8 ⋅ cos(30°)

= 8 ⋅ `sqrt3/2 = 4sqrt3`

= PR² = PQ² − QR²   ...[use Pythagoras for PR]

= `PR^2 = 8^2 − (4sqrt3)^2`

= PR² = 64 − 48

= PR² = 16

= PR = 4 cm

Step 3:

Perimeter = PQ + QR + PR

= `8 + 4sqrt3 + 4 = 4 + 4sqrt3` cm

= `(4 + 4sqrt3)` cm