Question

In the quadrilateral ABCD, AB = 40 cm, BC = 18 cm, CD = 24 cm, AD = 50 cm and ∠C = 90°. Find the (i) length of BD (ii) area of ABCD.

Answer 1

We are given quadrilateral ABCD with:

• AB = 40 cm
• BC = 18 cm
• CD = 24 cm
• AD = 50 cm
• ∠C = 90°

We are to find:

1. Length of diagonal BD
2. Area of quadrilateral ABCD

Step 1: Use Pythagoras in triangle ΔBCD

Since ∠C = 90°, triangle ΔBCD is right-angled at C.

Use: BD² = BC² + CD²

= 18² + 24²

= 324 + 576

= 900

⇒ `BD = sqrt(900)` 

⇒ BD = 30 cm

Step 2: Area of quadrilateral ABCD = Area(ΔABD) + Area(ΔBCD)

Area of ΔBCD right-angled at C:

Area_(ΔBCD) = `1/2` × BC × CD

= `1/2 xx 18 xx 24`

= 216 cm²

Area of ΔABD using Heron’s formula:

Sides: AB = 40, AD = 50, BD = 30

1. Semi-perimeter:

`s = (40 + 50 + 30)/2`

s = 60

2. Heron’s formula:

`"Area" = sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(60(60 - 40)(60 - 50)(60 - 30))`

= `sqrt(60 xx 20 xx 10 xx 30)`

= `sqrt(360000)`

= 600 cm²

Total Area of ABCD:

Area_(ABCD) = 600 + 216

Area_(ABCD) = 816 cm²