Question

A triangular piece of glass is being cut so that the height of the triangle is 4 inches shorter than twice the base. If the area of the triangle is 120 square inches, how long is the height of the triangle?

Answer 1

To find the length of the height of the triangle, we can set up two equations based on the given information: h = 2b - 4 and (1/2) * b * h = 120. By substituting the value of h from the first equation into the second equation and solving for b, we find that the base of the triangle is 12 inches. Substituting this value back into the first equation, we find that the height of the triangle is 20 inches.

Step-by-step explanation:

To find the length of the height of the triangle, let's first set up an equation. Let's denote the base of the triangle as 'b' and the height as 'h'. We are given that the height is 4 inches shorter than twice the base, so we can write the equation as: h = 2b - 4. We are also given that the area of the triangle is 120 square inches, so we can use the formula for the area of a triangle to set up another equation: (1/2) * b * h = 120. We can substitute the value of h from the first equation into the second equation to solve for b. Once we find the value of b, we can substitute it back into the first equation to find the value of h.

Let's solve the equations:

1. h = 2b - 4
2. (1/2) * b * h = 120

Using the first equation, we can substitute 2b - 4 for h in the second equation:

(1/2) * b * (2b - 4) = 120

Now, we can solve this equation for b. To do this, we can multiply both sides of the equation by 2:

b * (2b - 4) = 240

Expanding the left side, we get:

2b^2 - 4b = 240

Next, let's move all the terms to one side of the equation to make it a quadratic equation:

2b^2 - 4b - 240 = 0

Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula:

By factoring, we get: (2b + 20)(b - 12) = 0

Setting each factor equal to zero and solving for b, we find two possible values for b: b = -20/2 = -10 or b = 12.

Since the base cannot be negative, we discard b = -10 and conclude that the length of the base is b = 12 inches.

Now, we can substitute this value of b into the first equation to find the value of h:

h = 2(12) - 4 = 24 - 4 = 20 inches.

Therefore, the height of the triangle is 20 inches.