Triangle Classification Theorems

Assignment Active

Creating an Acute Triangle

The sides of a triangle have length x x + 4, and 20. If the length of the longest side is 20, which value of x would

make the triangle acute?

08

The value of x for the triangle to be acute triangle is 0 < x < 12.

Step-by-step explanation:

If we consider that a right triangle has side lengths x, x + 4, and 20 having 20 as the hypotenuse.

So, x² + (x + 4) ² = 20²

Now, the condition for the triangle with sides x, (x + 1), and 20 to be acute triangle is x² + (x + 4) ² < 20²

⇒ x² + x² + 8x + 16 < 400

⇒ 2x² + 8x - 384 < 0

⇒ x² + 4x - 192 < 0

⇒ x² + 16x - 12x - 192 < 0

⇒ (x + 16) (x - 12) < 0

Therefore, either (x + 16) > 0 and (x - 12) < 0

⇒ x > - 16 and x < 12

Or, (x + 16) 0

⇒ x 12, which is impossible.

Therefore, the value of x for the triangle to be an acute triangle is 0 < x < 12, as x can not be ≤ 0. (Answer)