Question

In ΔABC, AD ⊥ BC, AB = 60 cm, AD = 30 cm, and ∠C = 45°. Find

1. angles x
2. AC
3. BC

Answer 1

Given:

△ABC with:

AD ⊥ BC

AB = 60 cm

AD = 30 cm

∠C = 45°

Step 1:

Find angle x = ∠BAD

Triangle ABD,

`cos(x) = (AD)/(AB)`   ...[right angle at D]

cos(x) = `30/60`

cos(x) = `1/2`

x = `cos(-1) 1/2`

x = 60°

Step 2:

Find AC

∠C = 45°

∠DAC = 45°   ...[right-angled and remaining angle is 90° − 45°]

AD = DC     ...[AC is the hypotenuse of a 45° − 45° − 90° triangle.]

Hypotenuse = `"Leg" xx sqrt2`

AC = `30 xx sqrt2`

AC = 42.42

Step 3:

(iii) Find BC

BC = BD + DC

BD = `sqrt(AB^2 − AD^2)`

BD = `sqrt(60^2 − 30^2)`

BD = `sqrt(3600 − 900)`

BD = `sqrt2700`

BD = 51.96 cm

BC = 51.96 + 30   ...[since triangle ADC is isosceles and AD = DC]

BC = 81.96 cm