Question

the height of a triangle is 5 centimeters greater than the base. the area of the triangle is 63 square centimeters. find the length of the base and the height of the triangle.

Explanation:

Step1: Define variables

Let the base of the triangle be $x$ cm. Then the height is $(x + 5)$ cm.

Step2: Use the area formula

The area formula of a triangle is $A=\frac{1}{2}\times base\times height$. Given $A = 63$ square - centimeters, we have the equation $\frac{1}{2}\times x\times(x + 5)=63$.

Step3: Simplify the equation

Multiply both sides of the equation by 2 to get $x(x + 5)=126$. Expand the left - hand side: $x^{2}+5x=126$. Rearrange it to the standard quadratic form $x^{2}+5x - 126 = 0$.

Step4: Solve the quadratic equation

Factor the quadratic equation: $x^{2}+5x - 126=(x + 14)(x - 9)=0$. Set each factor equal to zero: $x+14 = 0$ gives $x=-14$; $x - 9=0$ gives $x = 9$. Since the length cannot be negative, we take $x = 9$.

Step5: Find the height

The height is $x + 5$. Substitute $x = 9$ into it, so the height is $9 + 5=14$ cm.

Answer:

Height = 14
Length of the base = 9