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Difficulty: HardTriangles and Pythagorean Theorem

An isosceles triangle has a perimeter of 3636 centimeters. The base of the triangle measures 1010 centimeters. What is the area of the triangle in square centimeters?

Answer: 60 square centimeters

Answer

60
To find the area, you must first know the perpendicular height. By splitting the isosceles triangle in half via its altitude, a right triangle is formed with a hypotenuse of 1313 (the side length) and a base of 55 (half of the full base). The Pythagorean theorem (52+h2=1325^2 + h^2 = 13^2) reveals the height is 1212. The full area is then calculated as 12×base×height\frac{1}{2} \times \text{base} \times \text{height}, which is 12×10×12=60\frac{1}{2} \times 10 \times 12 = 60.

Step-by-Step Solution

1
Calculate the length of the two equal sides.
The two equal sides each measure 1313 centimeters.
The perimeter is 3636 cm and the base is 1010 cm. The remaining 2626 cm must be divided equally between the two identical legs: 26÷2=1326 \div 2 = 13 cm.
2
Determine the base of the right triangle formed by the altitude.
The base of the right triangle is 55 centimeters.
The altitude of an isosceles triangle bisects the base, splitting the 1010-centimeter base into two equal 55-centimeter segments.
3
Calculate the height using the Pythagorean theorem.
The height is 1212 centimeters.
In the right triangle formed, the hypotenuse is 1313 and one leg is 55. Let hh be the height. 52+h2=132⇒25+h2=169⇒h2=144⇒h=125^2 + h^2 = 13^2 \Rightarrow 25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12.
4
Calculate the total area of the triangle.
The area is 6060 square centimeters.
The area formula is 12×base×height\frac{1}{2} \times \text{base} \times \text{height}. Thus, 12×10×12=60\frac{1}{2} \times 10 \times 12 = 60.

Key Concept

Using the Pythagorean theorem to find missing dimensions in isosceles triangles to calculate area.
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Topics

Triangles and Pythagorean Theorem

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