The problem is based on a right-angled triangle. The lengths of the two sides containing the right angle are given as $(x+2)$ cm and $(x+5)$ cm. The area of the triangle is $90$ cm². (i) Show that $x$ satisfies the equation $x^2 + 7x - 170 = 0$. (ii) Solve the equation to find the lengths of the sides containing the right angle.

1. Problem Description

The problem is based on a right-angled triangle. The lengths of the two sides containing the right angle are given as $(x+2)$ cm and $(x+5)$ cm. The area of the triangle is $90$ cm².

(i) Show that $x$ satisfies the equation $x^2 + 7x - 170 = 0$.

(ii) Solve the equation to find the lengths of the sides containing the right angle.

2. Solution Steps

(i) The area of a right-angled triangle is given by:

$Area = \frac{1}{2} \times base \times height$

In this case, the base is $(x+5)$ and the height is $(x+2)$. The area is given as $90$. Therefore, we have:

$\frac{1}{2} (x+5)(x+2) = 90$

Multiply both sides by 2:

$(x+5)(x+2) = 180$

Expand the left side:

$x^2 + 2x + 5x + 10 = 180$

$x^2 + 7x + 10 = 180$

Subtract 180 from both sides:

$x^2 + 7x - 170 = 0$

Hence, $x$ satisfies the equation $x^2 + 7x - 170 = 0$.

(ii) To solve the quadratic equation $x^2 + 7x - 170 = 0$, we can use the quadratic formula or factorization. Let's try factorization:

$x^2 + 7x - 170 = 0$

We need to find two numbers that multiply to -170 and add up to

7. Those numbers are 17 and -

1

0. $x^2 - 10x + 17x - 170 = 0$

$x(x - 10) + 17(x - 10) = 0$

$(x - 10)(x + 17) = 0$

Therefore, $x = 10$ or $x = -17$.

Since $x$ represents a length, it must be positive. Thus, $x = 10$.

Now, we can find the lengths of the sides:

Side 1: $x + 2 = 10 + 2 = 12$ cm

Side 2: $x + 5 = 10 + 5 = 15$ cm

3. Final Answer

(i) We have shown that $x$ satisfies the equation $x^2 + 7x - 170 = 0$.

(ii) The lengths of the sides containing the right angle are 12 cm and 15 cm.