The problem states that the height of an equilateral triangle is $(x+3)$ cm and the side length is 4 cm less than the height. (a) Express the area of the triangle, $L$ cm$^2$, in terms of $x$. (b) Given that the area of the triangle is 16 cm$^2$, find the height and side length of the triangle.

1. Problem Description

The problem states that the height of an equilateral triangle is $(x+3)$ cm and the side length is 4 cm less than the height.

(a) Express the area of the triangle, $L$ cm$^2$, in terms of $x$.

(b) Given that the area of the triangle is 16 cm$^2$, find the height and side length of the triangle.

2. Solution Steps

(a)

The height of the equilateral triangle is $h = x+3$.

The side length of the equilateral triangle is $s = h - 4 = (x+3) - 4 = x - 1$.

The area of an equilateral triangle can be expressed as:

$L = \frac{1}{2} \times base \times height = \frac{1}{2} \times s \times h$

Also, in an equilateral triangle, the height $h$ is related to the side $s$ by:

$h = \frac{\sqrt{3}}{2} s$

Thus, $x+3 = \frac{\sqrt{3}}{2} (x-1)$

However, we want to express L in terms of x. We know $s = x - 1$ and $h = x+3$. Thus the formula for the area of a triangle is

$L = \frac{1}{2} \times (x-1) \times (x+3)$

$L = \frac{1}{2} (x^2 + 3x - x - 3)$

$L = \frac{1}{2} (x^2 + 2x - 3)$

(b)

Given that the area is $16$ cm$^2$, we have $L = 16$.

$\frac{1}{2} (x^2 + 2x - 3) = 16$

$x^2 + 2x - 3 = 32$

$x^2 + 2x - 35 = 0$

$(x+7)(x-5) = 0$

Therefore $x = -7$ or $x = 5$.

Since the height and side length must be positive, we must have $x>1$. Thus $x = 5$.

The height $h = x+3 = 5+3 = 8$ cm.

The side length $s = x-1 = 5-1 = 4$ cm.

3. Final Answer

(a) $L = \frac{1}{2}(x^2 + 2x - 3)$

(b) Height $= 8$ cm, Side length $= 4$ cm