Calculating the Area of an Isosceles Triangle with Given Perimeters
Introduction
Isosceles triangles are a common topic in geometry and often required for various types of calculations. This article focuses on finding the area of an isosceles triangle when given its perimeter and the ratio between its base and the equal sides. We explore two methods: one using basic geometry and the Pythagorean theorem, and the other using Heron's formula. Each method will be detailed, ensuring a comprehensive understanding.Perimeter and Ratio
Given that the perimeter of an isosceles triangle is 32 cm and each side equal to its base is 5/6 of the base, we need to find the area of the triangle. Let's start by defining the variables and solving the problem step-by-step.Method 1: Basic Geometry and Pythagorean Theorem
Define the base and the equal sides: Let the base of the isosceles triangle be (x) cm. Then, each of the equal sides is (frac{5}{6}x) cm. Find the base using the perimeter:The perimeter of the isosceles triangle is given by:
[x frac{5}{6}x frac{5}{6}x 32]
Simplifying, we get:
[x frac{10}{6}x 32]
[frac{16}{6}x 32]
[frac{8}{3}x 32]
[8x 96]
[x 12,text{cm}]
Calculate the length of the equal sides:The equal sides are:
[frac{5}{6} times 12 10,text{cm}]
Find the altitude using the Pythagorean theorem:The altitude from the apex vertex to the base splits the isosceles triangle into two congruent right triangles. Applying the Pythagorean theorem:
[text{altitude}^2 10^2 - 12^2/4 100 - 36 64]
[text{altitude} sqrt{64} 8,text{cm}]
Calculate the area:The area of the isosceles triangle is:
[text{Area} frac{1}{2} times text{base} times text{altitude} frac{1}{2} times 12 times 8 48,text{cm}^2]
Method 2: Heron's Formula
Find the lengths of the sides:The base is (12,text{cm}) and each equal side is (10,text{cm}).
Calculate the semiperimeter:The semiperimeter (s) is:
[s frac{10 10 12}{2} 16,text{cm}]
Apply Heron's formula to find the area:Heron's formula states that the area (A) of a triangle is:
[A sqrt{s(s-a)(s-b)(s-c)}]
Here, (a 12,text{cm}), (b 10,text{cm}), and (c 10,text{cm}).
So,
[A sqrt{16(16 - 12)(16 - 10)(16 - 10)} sqrt{16 times 4 times 6 times 6} sqrt{4^2 times 6^2 times 2^2} 4 times 6 times 2 48,text{cm}^2]