Question

In the right △ABC, CD is the altitude to the hypotenuse AB and m∠ABC=30°. Find AD, if BD=54 cm.

Answer 1

The ratios of sides in a 30°-60°-90° triangle are such that the hypotenuse (AB) is double the length of the shortest side (AC). In your triangle, AC is the hypotenuse of similar triangle ACD, which has AD as its shortest side.

Then

... (1/2)×(AD+54) = AC

... (1/2)×AC = AD

Together, these tell you

... AD = (1/4)(AD +54)

... 3×AD = 54

... AD = 18

Answer 2

Answer:

18 cm

Step-by-step explanation:

Given: In the right $\Delta \text{ABC}$, CD is the altitude to the hypotenuse AB and $m\angle \text{ABC}=30^{\circ}$, $\text{BD}=54$ cm.

To find: AD

Solution: Consider the figure in the  attached file.

In $\Delta \text{BDC}$

$\text{cos\:30}^{\circ}=\frac{54}{BC}$

$\frac{\sqrt{3} }{2}=\frac{54}{BC}$

$\text{BC}=36\sqrt{3}$

Now, In $\Delta \text{ACB}$

$\text{cos\:30}^{\circ}=\frac{BC}{AB}$

$\frac{\sqrt{3} }{2}=\frac{36\sqrt{3}}{AB}$

$\text{AB}=36\sqrt{3}\times\frac{2}{\sqrt{3} }$

$\text{AB}=72$

Now, $\text{AB}=\text{AD}+\text{BD}$

$\text{AB}=\text{AD}+54$

$72=\text{AD}+54$

$\text{AD}=72-54$

$\text{AD}=18$ cm

Hence, $\text{AD}=18$ cm.