An isosceles triangle has length of sides |PQ|=10m, |PR|=13m and |QR|=13. Find the area of the triangle

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To find the area of the isosceles triangle, we first need to determine its height. We can do this by drawing an altitude from the apex to the base, which will divide the isosceles triangle into two congruent right-angled triangles.

Identify the base and equal sides.
The base of the triangle is $|PQ| = 10 \text{ m}$ .
The equal sides are $|PR| = 13 \text{ m}$ and $|QR| = 13 \text{ m}$ .
Find the height of the triangle.
When an altitude is drawn from the apex to the base of an isosceles triangle, it bisects the base. So, half of the base is $\frac{10 \text{ m}}{2} = 5 \text{ m}$ .
Now, we have a right-angled triangle with a hypotenuse of $13 \text{ m}$ and one leg of $5 \text{ m}$ . Let $h$ be the height (the other leg).
Using the Pythagorean theorem ( $a^2 + b^2 = c^2$ ):
$h^2 + (5 \text{ m})^2 = (13 \text{ m})^2$ $h^2 + 25 \text{ m}^2 = 169 \text{ m}^2$ $h^2 = 169 \text{ m}^2 - 25 \text{ m}^2$ $h^2 = 144 \text{ m}^2$ $h = \sqrt{144 \text{ m}^2}$ $h = 12 \text{ m}$
Calculate the area of the triangle.
The formula for the area of a triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ .
$\text{Area} = \frac{1}{2} \times 10 \text{ m} \times 12 \text{ m}$ $\text{Area} = 5 \text{ m} \times 12 \text{ m}$ $\text{Area} = 60 \text{ m}^2$
The area of the triangle is $\text{60 m}^2$ .

An isosceles triangle has length of sides |PQ|=10m, |PR|=13m and |QR|=13. Find the area of the triangle

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Step-by-step solution

Identify the base and equal sides.

Find the height of the triangle.

Calculate the area of the triangle.

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