The height of a triangle is 6 in. less than the length of the base. The area of the triangle is 80 inches squared, write an equation using x as your variable

Answer 1

The area of a triangle is 1/2 (base)(height).

Given that the height of the triangle is 6 less than the base you can write that mathematically as

h = b-6.

In the equation for the area of a triangle (A= 1/2*b*h), you can then replace the variable for height (h) with the equation that you created to represent the 'Given' part of the problem.

A = 1/2 (b) (h)

A = 1/2 (b) (b-6)

A = 1/2(b²-6b)

Because the question asks you to write an equation using x as your variable and only x, you can replace the variable b with x, understanding that the variable x represents the length of the base, b.

A = 1/2 (x² - 6x)

KYRA R. ASKED A FOLLOW UP QUESTION:

How would I solve for x in that equation?

My response:

Part of the problem indicates that the area of the triangle is 80 inches squared. Using the equation A = 1/2 (x² - 6x), I would substitute the variable A for area with the actual area, which is 80cm².

80cm² = 1/2 (x² - 6x)

To solve this quadratic equation, I would multiply both sides of the equation by 2.

2(80) = 2(1/2)(x² - 6x)

160 = x² - 6x

Then I would subtract 160 from both sides of the equation:

160-160 = x² - 6x -160

0 = x² - 6x -160

Now factor to solve for the values of x that make this equation equal to zero.

0 = (x-16)(x+10)

x-16 = 0 or x+10 = 0

x=16 or x = -10

Both values 16 and -10 are correct solutions if we were only solving this algebraic equation; however, the problem is a real world application that includes a triangle.

Recall from the beginning of the problem how I mentioned:

"understanding that the variable x represents the length of the base.'

The two solutions which represent the length of the base of the triangle are 16 and -10.

The base of the triangle has a length of 16 or -10; only one of these is correct. You cannot have a negative distance. For example, we could say that we drove 25 miles to the local animal shelter to adopt a puppy; we would NOT say that we drove negative 25 miles to the local animal shelter to adopt a puppy.

Thus the solution: x = 16; always check your solution if time permits.

A = 1/2 (b) (b-6)

80 = 1/2 (16) (16-6)

80 = (8) (10)

80 = 80 (true? if yes, your solution is correct)

Thank you Kyra! If you have any additional questions, please reach out to me.

All the best!