Question

a triangle has base ( 2x ) and height ( x - 3 ). the area of the triangle is 10 square inches.
what are the base and height of the triangle?
the base is 10 inches and the height is 1 inch.
the base is 2 inches and the height is 10 inches.
the base is 10 inches and the height is 2 inches.

Explanation:

Step1: Recall the area formula for a triangle

The area \( A \) of a triangle is given by \( A=\frac{1}{2}\times\text{base}\times\text{height} \). Here, base \( = 2x \) and height \( = x - 3 \), and \( A = 10 \). So we substitute these into the formula:
\[
10=\frac{1}{2}\times(2x)\times(x - 3)
\]

Step2: Simplify the equation

Simplify the right - hand side: \( \frac{1}{2}\times(2x)\times(x - 3)=x(x - 3)=x^{2}-3x \). So the equation becomes:
\[
x^{2}-3x = 10
\]
Rearrange it to a quadratic equation:
\[
x^{2}-3x - 10=0
\]

Step3: Solve the quadratic equation

Factor the quadratic equation. We need two numbers that multiply to \( - 10 \) and add up to \( - 3 \). The numbers are \( - 5 \) and \( 2 \). So,
\[
x^{2}-3x - 10=(x - 5)(x+2)=0
\]
Set each factor equal to zero:

• If \( x - 5=0 \), then \( x = 5 \).
• If \( x+2=0 \), then \( x=-2 \). But since \( x \) represents a length - related variable (base and height are lengths, so \( 2x>0 \) and \( x - 3>0 \)), \( x=-2 \) is not valid.

Step4: Find the base and height

When \( x = 5 \):

• Base \( = 2x=2\times5 = 10 \) inches.
• Height \( = x - 3=5 - 3 = 2 \) inches.

Answer:

The base is 10 inches and the height is 2 inches.