Question

the height of a triangle is 2 yards greater than the base. the area of the triangle is 144 square yards. find the length of the base and the height of the triangle. answer how to enter your answer (opens in new window) height = length of the base =

Explanation:

Step1: Define variables

Let the base of the triangle be $b$ yards. Then the height $h=b + 2$ yards.

Step2: Use area formula

The area formula of a triangle is $A=\frac{1}{2}bh$. Given $A = 144$ square - yards, we substitute $h=b + 2$ into the formula: $144=\frac{1}{2}b(b + 2)$.

Step3: Simplify the equation

Multiply both sides of the equation by 2 to get $288=b(b + 2)=b^{2}+2b$. Rearrange it to the quadratic - form $b^{2}+2b−288 = 0$.

Step4: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b = 2$, $c=-288$), we use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(2)^{2}-4\times1\times(-288)=4 + 1152 = 1156$. Then $b=\frac{-2\pm\sqrt{1156}}{2}=\frac{-2\pm34}{2}$. We have two solutions for $b$: $b_1=\frac{-2 + 34}{2}=\frac{32}{2}=16$ and $b_2=\frac{-2-34}{2}=\frac{-36}{2}=-18$. Since the length of the base cannot be negative, we take $b = 16$ yards.

Step5: Find the height

Since $h=b + 2$, when $b = 16$, then $h=16 + 2=18$ yards.

Answer:

Height = 18 yards
Length of the base = 16 yards