Question

the height of a triangle is 3 feet less than twice its base. the area of the triangle is 52 ft². what is the base of the triangle? ___ ft enter the answer check it

Explanation:

Step1: Define variables

Let the base of the triangle be $b$ feet. Then the height $h = 2b - 3$ feet.

Step2: Use area formula

The area formula of a triangle is $A=\frac{1}{2}bh$. Given $A = 52$ square - feet. Substitute $h = 2b - 3$ and $A=52$ into the formula:
$52=\frac{1}{2}b(2b - 3)$

Step3: Simplify the equation

Multiply both sides by 2 to get rid of the fraction: $104=b(2b - 3)$. Expand the right - hand side: $104 = 2b^{2}-3b$. Rearrange to get a quadratic equation: $2b^{2}-3b - 104 = 0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 2$, $b=-3$, $c=-104$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ or factor. Factoring $2b^{2}-3b - 104=(2b + 13)(b - 8)=0$.
Set each factor equal to zero:
$2b+13 = 0$ gives $b=-\frac{13}{2}$; $b - 8=0$ gives $b = 8$.
Since the base of a triangle cannot be negative, we discard $b=-\frac{13}{2}$.

Answer:

8