The altitude drawn to the base of an isosceles triangle is 16 cm and its perimeter is 64 cm. What is the area of the triangle?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 192 cm2
In an isosceles triangle, the altitude to the base cuts the base into two equal halves, so it splits the triangle into two equal right triangles. Let each equal side be \(a\) cm and half the base be \(b\) cm.
The two sides (from the perimeter and from Pythagoras):
\(2a+2b=64\), so \(a+b=32\) (1)
\(a^{2}-b^{2}=16^{2}=256\)
\((a+b)(a-b)=256\), so \(32(a-b)=256\) and \(a-b=8\) (2)
Adding (1) and (2), the \(b\) terms cancel: \(2a=40\), so \(a=20\) and \(b=12\)
The area:
Base \(=2b=24\) cm
Area \(\displaystyle =\frac{1}{2}\times24\times16=\) 192 cm2