Solving for the Base of a Triangle Given Area and Relationship to Height

Q21. A triangle is 8 cm wider than it is tall. The area is 280 cm2. Find the width (length of the base).

Background

Topic: Solving quadratic equations and geometric applications.

This question tests your ability to set up and solve a quadratic equation based on the formula for the area of a triangle, using a relationship between the base and height.

Key Terms and Formulas

• Area of a triangle: $A = \frac{1}{2} \times \text{base} \times \text{height}$

• Let $h$ be the height of the triangle (in cm).

• The base is $h + 8$ (since it is 8 cm wider than the height).

• The area is given as $280$ cm2.

Step-by-Step Guidance

1. Write the area formula for the triangle using the given values: $A = \frac{1}{2} \times (h + 8) \times h$.

2. Set the area equal to $280$cm2: $280 = \frac{1}{2} \times (h + 8) \times h$.

3. Multiply both sides by $2$to clear the fraction: $560 = (h + 8) \times h$.

4. Expand the right side: $560 = h^2 + 8h$.

5. Rearrange the equation to standard quadratic form: $h^2 + 8h - 560 = 0$.

Try solving on your own before revealing the answer!

Final Answer: 28 cm

Solving the quadratic equation $h^2 + 8h - 560 = 0$ gives $h = 20$. The base is $h + 8 = 28$ cm.

This matches the relationship given in the problem and the area calculation.