Question

The sides of a triangle have lengths 2x,8, and 12 . If the length of the longest side is 2x, what values of x make the triangle acute

Answer 1

The values of x that make the triangle acute are any values of x greater than the square root of 20.

Step-by-step explanation:

In order for a triangle to be acute, the sum of the squares of the two shorter sides must be greater than the square of the longest side. In this case, the longest side is 2x, so we need to compare the sum of the squares of 2x and 8 with the square of 12.

(2x)^2 + 8^2 > 12^2

Simplifying the equation, we get:

4x^2 + 64 > 144

Subtracting 64 from both sides:

4x^2 > 80

Dividing both sides by 4:

x^2 > 20

Taking the square root of both sides:

x > √20

So, the values of x that make the triangle acute are any values of x greater than the square root of 20.