The height of a triangle is 9 m less than it’s base. The area of the triangle is 56m 2. Find the length of the base

Hi there! The answer is 16 meters.

To find our answer, we must set up and solve an equation.

First step: Set up an expression
Let the base of the triangle be represented by x.
Now we can conclude that the heigth of the triangle can be expressed by the expression x - 9 (since the height is 9m less than it's base).

Second step: Find the area, expressed in terms of x.
The area of a triangle can be found using the following formula
A = base * height * 0.5
(or A = (base * height) / 2)
When we plug in our data, we can find an expression of the area. We express the area in terms of x.

[tex]A = 0.5x(x-9)[/tex]

Step 3: Simplify
[tex]A = 0.5x(x-9)[/tex]
Work out the parenthesis using rainbow method.

[tex]A = 0.5x^{2} - 4.5x[/tex]

Step 4: Set up and solve the equation
When we've found our area expressed in terms of x, we can set up an equation.

[tex]0.5 x^{2} -4.5x = 56[/tex]
Subtract 56

[tex]0.5 x^{2} -4.5 - 56=0[/tex]
Divide by 2.

[tex] x^{2} - 9 x- 112 = 0 [/tex]
[tex](x+7)(x-16) = 0[/tex]
Rule AB = 0, gives A = 0 or B = 0

[tex]x + 7 = 0 \\ x = -7\\ \\ x - 16 = 0\\ x = 16 [/tex]

Step 5: Conclusion
The length of the base of the triangle (which was represented by x) is 16 metres (it can't be -7, since that's a negative number and length can't be negative).

The answer is 16 meters.
~ Hope this helps you!