Question : $\triangle ABC$ is an isosceles triangle with AB = AC = 15 cm and an altitude from A to BC of 12 cm. The length of side BC is:
Option 1: 9 cm
Option 2: 12 cm
Option 3: 18 cm
Option 4: 20 cm
Correct Answer: 18 cm
Solution :
Given: $\triangle ABC$ is an isosceles triangle with $AB = AC = 15$ cm.
$AD \perp BC$
So, $AD = 12$ cm.
Also, $BD = DC$.
Using Pythagoras theorem in $\triangle ABD$,
$(BD)^2=(AB)^2–(AD)^2$
$⇒BD=\sqrt{(AB)^2–(AD)^2}$
$⇒BD=\sqrt{(15)^2–(12)^2}$
$⇒BD=\sqrt{(15–12)(15+12)}$
$⇒BD=\sqrt{3\times27}$
$⇒BD=9$ cm
$BC = 2 \times BD = 2 \times 9 =18$ cm.
So, the length of the side BC is 18 cm.
Hence, the correct answer is 18 cm.
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