Question

the area of a triangle is 36 cm². the height of the triangle is 6 cm less than the base. what is the height of the triangle?

Explanation:

Step1: Let the base be $x$ cm.

The height is $(x - 6)$ cm.

Step2: Use the triangle - area formula.

The area formula of a triangle is $A=\frac{1}{2}\times base\times height$. Given $A = 36$ cm², we have the equation $\frac{1}{2}\times x\times(x - 6)=36$.

Step3: Simplify the equation.

Multiply both sides of the equation by 2 to get $x(x - 6)=72$. Expand the left - hand side: $x^{2}-6x=72$. Rearrange it to the standard quadratic form $x^{2}-6x - 72 = 0$.

Step4: Solve the quadratic equation.

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b=-6$, $c=-72$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor. Factoring $x^{2}-6x - 72=(x - 12)(x + 6)=0$. So $x=12$ or $x=-6$. Since the length cannot be negative, the base $x = 12$ cm.

Step5: Find the height.

The height is $x-6$. Substitute $x = 12$ into it, and the height is $12-6=6$ cm.

Answer:

6 cm