Question

In ΔABE, C and D are points on EB. ∠B = 90°, BC = 5 cm, CD = 4 cm and DE = 26 cm. Area of ΔADC = 24 cm². Find the length of x, y and z.

Answer 1

Given:

• Triangle ΔABE with ∠B = 90°
• Points C and D lie on EB such that BC = 5 cm, CD = 4 cm and DE = 26 cm
• Area of ΔADC = 24 cm²
• The diagram shows lengths x = AB, y = AD and z = AE which need to be found.

Stepwise Calculation:

1. Since ∠B = 90°, ΔABE is a right-angled triangle at B.

2. Since C and D lie on EB:

EB = BC + CD + DE 

= 5 + 4 + 26

= 35 cm

3. In ΔADC, base DC = 4 cm segment on EB between points D and C.

4. Area (ΔADC) = 24 cm². 

`"Area" = 1/2 xx "base" xx "height"`

Height from A perpendicular to DC is x the height from A to EB line.

Thus, `24 = 1/2 xx 4 xx "height"`

⇒ Height = `(24 xx 2)/4`

⇒ Height = 12 cm

5. This height of 12 cm corresponds to AB (x), the perpendicular side of the right triangle at B:

So x = AB = 12 cm

6. To find y = AD:

AD is hypotenuse of right triangle ABD where BD

= BC + CD 

= 5 + 4

= 9 cm

Using Pythagoras theorem:

AD² = AB² + BD²

= 12² + 9²

= 144 + 81

= 225

`AD = sqrt(225)`

AD = 15 cm

⇒ y = 15 cm

7. To find z = AE:

AE is hypotenuse of right triangle ABE where BE = 35 cm and AB = 12 cm.

AE² = AB² + BE²

= 12² + 35²

= 144 + 1225

= 1369

`AE = sqrt(1369)`

AE = 37 cm

⇒ z = 37 cm

x = AB = 12 cm

y = AD = 15 cm

z = AE = 37 cm