Question

The adjacent sides of a ||gm ABCD measure 34 cm and 20 cm and the diagonal AC is 42 cm long. Find the area of the ||gm.

Answer 1

Given: Adjacent sides AB = 34 cm, AD = 20 cm, diagonal AC = 42 cm.

Step-wise calculation:

1. Let the angle between the adjacent sides (∠BAD) be θ.

For the diagonal AC (vector sum of AB and AD):

AC² = AB² + AD² + 2 × AB × AD × cos θ.

2. Plug numbers: 42² = 34² + 20² + 2 × 34 × 20 × cos θ 

⇒ 1764 = 1156 + 400 + 1360 × cos θ 

⇒ 1764 – 1556 = 1360 × cos θ 

⇒ 208 = 1360 × cos θ 

⇒ cos θ = `208/1360`

⇒ cos θ = `13/85`

3. Then `sin θ = sqrt(1 - cos^2θ)`

= `sqrt(1 - (13/85)^2)`

= `sqrt((7225 - 169)/7225)` 

= `sqrt(7056/7225)`

= `84/85`

4. Area of parallelogram = AB × AD × sin θ 

= `34 xx 20 xx (84/85)`

= `680 xx (84/85)`

= 8 × 84 

= 672 cm²