Word Problems - Area of Triangles

Triangle Word Problems (Quadratic Equations)
Possible scenarios that can lead to quadratic equations:

1. Dimensions Expressed Linearly with a Variable: If both the base and the height of a triangle are given as linear expressions involving the same variable (e.g., base = \(x + 2\), height = \(2x - 1\)), and the area is known, setting up the area formula will result in a quadratic equation.
2. Relationships Between Dimensions: The problem might state a relationship between the base and height (or other dimensions) that, when combined with the area formula, leads to a quadratic equation.

The following diagram gives an example of a triangle word problem solved using a quadratic equation. Scroll down the page for more examples and solutions of triangle word problems.

 

Algebra Worksheets
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Printable & Online Algebra Worksheets

Example:
A right-angled triangle has legs with lengths \(x\) and \(x + 7\). If the area of the triangle is 30 square units, find the lengths of the legs.
Solution:
In a right-angled triangle, the legs can be the base and height.
Base = \(x\)
Height = \(x + 7\)
Area = \(\frac{1}{2} \cdot \text{base} \cdot \text{height}\)
\(30 = \frac{1}{2} \cdot x \cdot (x + 7)\)
Multiply by 2: \(60 = x(x + 7)\)
Expand: \(60 = x^2 + 7x\)
Rearrange into a quadratic equation: \(x^2 + 7x - 60 = 0\)
Factor the quadratic: \((x + 12)(x - 5) = 0\)
Since the length cannot be negative, we take \(x = 5\).
The lengths of the legs are:
Base = \(x = 5\) units
Height = \(x + 7 = 5 + 7 = 12\) units

Answer: The lengths of the legs are 5 units and 12 units.
Check: Area = \(\frac{1}{2} \cdot 5 \cdot 12 = 30\) (Correct).

Writing quadratic equations to solve word problems: Area of a triangle
Example:
The height of a triangle is 3 cm more than the base. The area of the triangle is 17 cm². Find the base to the nearest hundredth of a cm.

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How to solve a triangle if you know the area?
This is a word problem you solve by writing an equation and factoring.
Word Problem: Area of a Triangle

Example:
The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. Find the height of the triangle.

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Area of a Triangle - Algebra and Geometry Help
Example:
The base of triangle is 5 units less than twice the height. If the area is 75 square units, then what is the length of the base and the height?

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Algebra and Triangles : solving equations linked to perimeter and area
Example:
The sides of a triangle are given as 3x, x - 1 and 3x + 1. If the perimeter is 56m, find the area.

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Find area of a triangle- word problem
Example:
In triangle ABC, AD = 10cm, the length of CD is half the length of AD, and the length of BD is twice the length of AD. What is the area of triangle ABC?

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How to find the dimensions of a triangle given its area?
This problem would involving solving quadratic equations.
Example:
The height of a triangle is 4 inches less than the length of the base. The area of the triangle is 30 in². Find the height and the base.

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Word problem involving area of triangle
Example:
Before she goes camping, La Verne has to buy a tent pole to replace the one she lost on her last outing. If the area of the front of the tent is 22 square feet and the base of the tent has the dimensions indicated below, how tall must the pole be?

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