Answer:
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 and (x - 12) > 0
⇒ x < - 16 and 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)
