Question

The lengths of two sides of a right triangle containing the right angle differ by 6 cm. If the area of the triangle is 36 cm^2, find the perimeter of the triangle.

Answer 1

Question #44043 – Math – Geometry

The lengths of two sides of a right triangle containing the right angle differ by $6\, \text{cm}$. If the area of the triangle is $36\, \text{cm}^2$, find the perimeter of the triangle.

Solution:

Let $x$ is the length of $AC$, then $CB = x + 6$ is the length of a base.

The area of the triangle is:

$$S = \frac{AC \cdot BC}{2} = \frac{x(x + 6)}{2}$$

Given $S = 36$.

$$\frac{x(x + 6)}{2} = 36$$

Expanding:

$$x^2 + 6x - 72 = 0$$ $$x = -3 \pm \sqrt{9 + 72} = -3 \pm 9$$ $$x > 0, \text{ so } x = 6$$ $$AC = 6\, \text{cm}$$ $$CB = x + 6 = 6 + 6 = 12\, \text{cm}$$

Using Pythagoras' theorem:

$$AC^2 + CB^2 = AB^2$$ $$AB = \sqrt{AC^2 + CB^2} = \sqrt{12^2 + 6^2} \approx 13$$

The perimeter of the triangle:

$$P = AB + AC + CB = 6 + 12 + 13 = 31\, \text{cm}$$

www.AssignmentExpert.com