Algebra: Geometry Word Problems - Area
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How to solve geometry word problems that involve geometric figures and angles described in words?
Algebra word problems involving area typically require setting up and solving equations based on the formulas for the area of geometric shapes (e.g., rectangles, triangles, circles). These problems often involve finding unknown dimensions or areas using algebraic methods.
Geometry Word Problems Involving Area
Key Steps for Solving Area Word Problems:
- Identify the shape and its properties. Making a sketch of the geometric figure is often helpful.
- Write the formula for the area of the shape.
- Set up equations based on the given information.
- Solve the equations to find the unknown variables.
- Check the solution to ensure it makes sense in the context of the problem.
Here are some examples of algebra word problems involving area, along with step-by-step solutions:
The following diagram gives the steps to solve an area word problem using Algebra.
Geometry Word Problems
Perimeter Word Problems
Area Word Problems
Volume Word Problems
Angle Word Problems
Example:
A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?
Solution:
Step 1: Assign variables:
Let x = original width of rectangle
Sketch the figure
Step 2: Write out the formula for area of rectangle.
A = lw
Step 3: Plug in the values from the question and from the sketch.
60 = (4x + 4)(x –1)
Use distributive property to remove brackets
60 = 4x2 – 4x + 4x – 4
Put in Quadratic Form4x2 – 4 – 60 = 0
4x2 – 64 = 0
This quadratic can be rewritten as a difference of two squares
(2x)2 – (8)2 = 0
Factorize difference of two squares
(2x)2 – (8)2 = 0
(2x – 8)(2x + 8) = 0
We get two values for x.
2x - 8 = 0 ⇒ 2x = 8 ⇒ x = 4
2x + 8 = 0 ⇒ 2x = -8 ⇒ x = -4
Since x is a dimension, it would be positive. So, we take x = 4
The question requires the dimensions of the original rectangle.
The width of the original rectangle is 4.
The length is 4 times the width = 4 × 4 = 16
Answer: The dimensions of the original rectangle are 4 and 16.
Writing quadratic equations to solve word problems: Area of a triangle
Example:
The height of a triangles is 3 cm more than its base. The area of the triangle is 17 cm2. Find the base to nearest hundredth of a cm.
Find the Dimensions of a Rectangle Word Problem
Example:
The length of a rectangle is 5 units more than twice its width. If the area is 250 sq. units. then find the dimensions of the rectangle.
Solve Area World Problems by Factoring
Example:
A garden that is 4 meters wide and 6 meters long is to have a uniform border such that the area of the border is the same as the area of the garden. Find the width of the border?
Example of geometry word problem that involves area
Example:
A rectangle is twice as long as it is wide. If the area of the rectangle is 98 cm2, find its dimensions.
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