Question

The altitude drawn to the base of an isosceles triangle is 8cm and the perimeter is 32cm. Find the area of the triangle?

• A. 48
• B. 60
• C. 70
• D. 80

Hint:

Always draw a diagram for geometry problems. Use algebraic equations for lengths and apply theorems like Pythagoras. Double-check calculations and compare with options. If discrepancies arise, highlight them.

Answer 1

The Correct Option is A

Step 1: Let the half-base be \( x \) and each equal side be \( y \). The perimeter gives \( 2x + 2y = 32 \), so \( x + y = 16 \).

Step 2: The altitude of 8 cm and Pythagoras give \( y^2 - x^2 = 8^2 = 64 \). Since \( y^2 - x^2 = (y-x)(y+x) \), and \( y+x=16 \), this means \( y - x = \frac{64}{16} = 4 \).

Step 3: Solving \( x+y=16 \) and \( y-x=4 \) together gives \( y=10 \) and \( x=6 \), so the base is \( 2x = 12 \) cm.

Step 4: Area \( = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 8 = 48 \).
\[ \boxed{48} \]