Question

Using the altitude theorem. I have a triangle, the altitude is 20. a is 25 but there's no b. Just the sum of a and b which is x. How do I solve this problem?

Answer 1

Explanation:

We're given a right triangle with an altitude h = 20 that divides the hypotenuse into two segments a and b, and given a = 25 and a + b = x.

The altitude theorem states that h = sqrt(a*b), or equivalently that h^2 = a*b. We already know what a and h are; now let's solve for b.

a + b = x

a + b - a = x - a

b = x - a

So then

h^2 = a*b

h^2 = a*(x - a)

h^2 = ax - a^2

20^2 = 25x - 25^2

400 = 25x - 625

400 + 625 = 25x - 625 + 625

1025 = 25x

1025/25 = 25x/25

x = 41

Now we plug this back into our equation for b.

b = x - a

b = 41.- 25

b = 16

We can verify that this is the correct answer using the altitude theorem.

h^2 = a*b

20^2 = 25*16

400 = 400