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Find the area of the triangle

Question: Two sides of a triangle are 5 and 9, the third side is the root of the equation x2 - 8x + 15 = 0. What is the area of the triangle?

Solution:

First, we need to find the length of the third side. The length of the third side satisfy the equation x2 - 8x + 15 = 0. Now we factoring the left-side of the equation, (x - 3) (x - 5) = 0, so x1 = 3, and x2 = 5. Because for any triangle, the sum of any two sides is longer than the third side. If the third side is 3, then 3 + 5 = 8, 8 < 9, so 3 is not a side in this triangle. If the third side is 5, then 5 + 5 = 10, 10 > 9, so the third side is 5.

Because the two sides of the triangle are equal, so the triangle is an isosceles triangle.

For an isosceles triangle, its altitude to the base cuts the base in half.

h2 = 52 - (9/2)2 = 25 - 81/4 = 100/4 - 81/4 = 19/4
h = (square root of 19)/2
The area (A) of the triangle,
A = (1/2) base × altitude = (1/2) 9 × (square root of 19)/2 = 9 (square root of 19)/4.