Math Problem Statement
Solution
To solve this problem, let's analyze the equations and the geometry of the rectangle:
Problem Breakdown:
-
Given:
- ABCD is a rectangle.
- AC (one diagonal) is expressed as .
- BD (the other diagonal) is expressed as .
-
Key Property of a Rectangle:
- In a rectangle, the diagonals are equal in length. Hence, .
-
Set the equations equal:
Solution:
Step 1: Eliminate the fraction by multiplying through by 4:
Step 2: Simplify:
Step 3: Rearrange to isolate :
Step 4: Solve for :
Step 5: Calculate the lengths of AC and BD:
- Substitute into the expressions for and :
- .
- .
Thus, the length of each diagonal is .
Would you like a detailed step-by-step explanation of any part? Let me know if you have questions or need clarification!
Related Questions:
- How can we use the property of diagonals to determine unknown variables in other quadrilaterals?
- If one diagonal's formula includes a square term, how would the solution process change?
- What are some real-life applications of rectangle diagonal properties?
- How would the problem change if ABCD were a square instead of a rectangle?
- Can you solve for if one diagonal was twice the other?
Tip: Always check the geometry properties of the shape when solving problems involving diagonals, sides, or angles.
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Math Problem Analysis
Mathematical Concepts
Algebra
Properties of Rectangles
Equations
Formulas
Diagonal lengths are equal in a rectangle
Theorems
Diagonal equality property of rectangles
Suitable Grade Level
Grades 7-10
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