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------------------------------------------------ Find the height of the triangle by applying formulas for the area of a triangle and your knowledge about triangles.

This is a triangle with:
- Side a: 9 inches
- Side b: 9 inches
- Side c: 6 inches

The altitude to side c has a length of x inches.

A. 11.3 in
B. 8.5 in
C. 6.2 in
D. 8 in

Question

High School

Answer :

akibkhanzai12 akibkhanzai12 · Answer 1 · 2024-08-01T00:19:30-04:00

The height of the triangle is approximately 8.48 inches.

Among the given answer choices, the closest option is (b) 8.5 inches.

To find the height of the triangle, we can use the formula for the area of a triangle:

Area = (1/2) base x height

In this case, side c serves as the base, and the altitude (height) to side c is given as x inches. Therefore, we can write:

Area = (1/2) 6 x

Now, let's calculate the area of the triangle using Heron's formula:

s = (a + b + c)/2 = (9 + 9 + 6)/2 = 12

Area = sqrt(s (s - a) (s - b) (s - c)

= sqrt(12 (12 - 9) (12 - 9) (12 - 6)

= sqrt(12 3 3 6)

= sqrt(648)

= 18√2

Now, equating the two expressions for the area, we have:

(1/2) 6 x = 18√2

Simplifying the equation, we get:

3x = 18√2

Dividing both sides by 3, we have:

x = 6√2

Approximating the value, x ≈ 6 1.414 = 8.48

Therefore, the height of the triangle is approximately 8.48 inches.

Among the given answer choices, the closest option is (b) 8.5 inches.

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