Question

in the right △abc, $overline{cd}$ is the altitude to the hypotenuse $overline{ab}$ and m∠abc = 30°. find ad if bd = 54 cm. answer: ad = cm

Explanation:

Step1: Use geometric - mean theorem

In right - triangle $ABC$ with altitude $CD$ to hypotenuse $AB$, we have $CD^{2}=AD\times BD$. Also, in right - triangle $BCD$, $\cos\angle ABC=\frac{BD}{BC}$. Since $\angle ABC = 30^{\circ}$ and $BD = 54$ cm, $BC=\frac{BD}{\cos30^{\circ}}=\frac{54}{\frac{\sqrt{3}}{2}} = 36\sqrt{3}$ cm.

Step2: Use similarity of triangles

Triangles $ACD$ and $BCD$ are similar to triangle $ABC$. In right - triangle $ABC$, $\cos\angle ABC=\frac{BC}{AB}$. Let $AD=x$. Then $AB=x + 54$. Also, from the property of right - triangle with a $30^{\circ}$ angle, $BC^{2}=BD\times AB$. Substitute $BC = 36\sqrt{3}$ and $BD = 54$ into $BC^{2}=BD\times AB$: $(36\sqrt{3})^{2}=54\times(x + 54)$.

Step3: Solve for $AD$

$3888=54x+2916$. Then $54x=3888 - 2916=972$. So $x = 18$ cm.

Answer:

$18$