Question

the height of a triangle is 5 meters greater than the base. the area of the triangle is 150 square meters. find the length of the base and the height of the triangle.

Explanation:

Define variables and set up the equation

Let \(b\) represent the length of the base of the triangle in meters.
The height \(h\) is \(5\) meters greater than the base:

$$h = b + 5$$

The area of a triangle is given by:

$$A = \frac{1}{2} \cdot \text{base} \cdot \text{height}$$

Substitute the given area \(150\text{ m}^2\) and the expression for height:

$$150 = \frac{1}{2} b (b + 5)$$

Formulate the quadratic equation

Multiply both sides by \(2\) to clear the fraction:

$$300 = b(b + 5)$$

Expand the right side:

$$300 = b^2 + 5b$$

Subtract \(300\) from both sides to write the equation in standard form:

$$b^2 + 5b - 300 = 0$$

Solve the quadratic equation by factoring

Using the Factoring Quadratic Equations knowledge point

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Find the valid physical dimensions

Using the Solving Quadratic Equations knowledge point

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Calculate the height

The base length is \(b = 15\text{ m}\).
Calculate the height \(h\):

$$h = b + 5 = 15 + 5 = 20\text{ m}$$

Answer:

The height of a triangle is 5 meters greater than the base. The area of the triangle is 150 square meters. Find the length of the base and the height of the triangle.

Height = <blank>20</blank> meters
Length of the base = <blank>15</blank> meters