Math Problem Statement
Which irrational number represents the length of each side of the squares? The area of the rectangle is 90. You can also cut it into 2 squares
Solution
Let's break down the problem:
- Given: The area of the rectangle is 90, and it can be cut into two squares.
- Question: We need to find the length of each side of the squares, represented by an irrational number.
Solution:
Let be the length of the side of each square.
- The area of one square is .
- Since the rectangle is cut into two squares, the total area of the rectangle will be the sum of the areas of the two squares, i.e., .
Given that the area of the rectangle is 90:
Dividing both sides by 2:
Taking the square root of both sides:
Simplifying :
Thus, the length of each side of the squares is , which is an irrational number.
Final Answer:
The length of each side of the squares is , which is an irrational number.
Would you like further details or explanations? Here are five related questions to think about:
- How do we know that is irrational?
- Can we approximate to a decimal value?
- How would the solution change if the rectangle were cut into more than two squares?
- What are other examples of irrational numbers that can represent lengths in geometric problems?
- How do you determine if a square root simplifies to a rational number?
Tip: When simplifying square roots, always factor the number into its prime components to check if there are perfect squares you can simplify.
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Math Problem Analysis
Mathematical Concepts
Geometry
Algebra
Irrational Numbers
Formulas
Area of a rectangle
Area of a square
Simplifying square roots
Theorems
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Suitable Grade Level
Grades 9-12
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