Question

In the quadrilateral ABCD, ∠ACB = 90°, AB = 20 cm, BC = 12 cm and AD = CD = 17 cm. Find (i) AC (ii) area of quadrilateral ABCD.

Answer 1

Given:

• Quadrilateral ABCD with ∠ACB = 90°
• AB = 20 cm
• BC = 12 cm
• AD = CD = 17 cm

We need to find:

1. AC 
2. Area of quadrilateral ABCD

Step 1: Find AC using right triangle ABC

Since ∠ACB = 90°, triangle ABC is a right-angled triangle at C.

Use Pythagoras theorem: AB² = AC² + BC² 

Substitute known values: 20² = AC² + 12²
400 = AC² + 144

AC² = 256

AC = `sqrt(256)`

AC = 16 cm

Step 2: Find the area of quadrilateral ABCD

Quadrilateral ABCD can be divided into two triangles:

Triangle ABC   ...(Right-angled at C)

Triangle ADC   ...(Isosceles triangle with AD = CD = 17 cm)

Area of triangle ABC:

`"Area" = 1/2 xx AC xx BC`

= `1/2 xx 16 xx 12`

= 96 cm²

Find the area of triangle ADC:

Use Heron’s formula: 

`s = (AD + DC + AC)/2`

= `(17 + 17 + 16)/2`

= 25 cm

`Area = sqrt(s(s - AD)(s - DC)(s - AC))`

= `sqrt(25(25 - 17)(25 - 17)(25 - 16))`

= `sqrt(25 xx 8 xx 8 xx 9)`

= `sqrt(25 xx 576)`

= `sqrt(14400)`

= 120 cm²

Total area of quadrilateral ABCD:

= Area(ΔABC) + Area(ΔADC)

= 96 + 120

= 216 cm²

AC = 16 cm

Area of quadrilateral ABCD = 216 cm²