Question

In ΔPQR, ∠Q = 90° and PQ = QR = 6 cm. Calculate the

1. area of triangle
2. length of perpendicular from Q to PR  [Take `sqrt2 = 1.414`]

Answer 1

Given:

• In ΔPQR, ∠Q = 90°
• PQ = QR = 6 cm
• Take `sqrt2` = 1.414

Stepwise calculation:

Step 1: Identify triangle type and sides

Since ∠Q = 90° and PQ = QR = 6 cm, triangle PQR is an isosceles right triangle with legs PQ and QR equal.

Step 2: Calculate the area of ΔPQR

Area of a right triangle = `1/2 xx "leg"_1 xx "leg"_2`

`"Area" = 1/2 xx  PQ xx QR`

= `1/2 xx 6 xx 6`

= 18 cm²

Step 3: Calculate length of hypotenuse PR

Using Pythagoras theorem:

`PR = sqrt(PQ^2 + QR^2)`

= `sqrt(6^2 + 6^2)`

= `sqrt(72)`

= `6sqrt(2)`

= 6 × 1.414

= 8.484 cm

Step 4: Calculate the length of perpendicular from Q to PR

Since ∠Q = 90°, Q lies on the right angle vertex, the perpendicular from Q to PR is actually the height in the triangle relative to hypotenuse PR.

The length of the perpendicular height from Q to PR in a right triangle is given by:

`"Height" = ("Area" xx 2)/("Base")`

= `(2 xx 18)/8.484`

= `36/8.484 ≈ 4.24  cm`

Area of triangle ΔPQR = 18 cm².

Length of perpendicular from Q to PR = 4.24 cm.