Question

the height of a triangle is 7 centimeters greater than the base. the area of the triangle is 39 square centimeters. find the length of the base and the height of the triangle.

Explanation:

Define variables and set up the equation

Using the Geometric Quadratic Modeling knowledge point
Let \(b\) represent the base of the triangle in centimeters.
The height \(h\) is \(7\text{ cm}\) greater than the base:

$$h = b + 7$$

The area of a triangle is given by:

$$A = \frac{1}{2} \cdot b \cdot h$$

Substitute the given area \(39\text{ cm}^2\) and the expression for \(h\):

$$39 = \frac{1}{2} b(b + 7)$$

Write the quadratic equation in standard form

Using the Standard Form of Quadratic Equations knowledge point
Multiply both sides by 2 to clear the fraction:

$$78 = b(b + 7)$$

$$78 = b^2 + 7b$$

$$b^2 + 7b - 78 = 0$$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point
Factor the quadratic equation by finding two numbers that multiply to \(-78\) and add to \(7\). These numbers are \(13\) and \(-6\):

$$(b + 13)(b - 6) = 0$$

Set each factor to zero:

$$b + 13 = 0 \implies b = -13$$

$$b - 6 = 0 \implies b = 6$$

Determine the physical dimensions

Since a length must be positive, we discard the negative solution \(b = -13\).
Therefore, the base is:

$$b = 6\text{ cm}$$

The height is:

$$h = b + 7 = 6 + 7 = 13\text{ cm}$$

Answer:

Height = 13
Length of the base = 6