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Solve for x and y Using Square Diagonal Angle Properties
Mathematics
Grade 8 (Junior High School)
Question Content
In square ABCD with diagonals intersecting at O, ∠CBD = 7x - 11 and ∠AOD = 5y + 30. Solve for x and y.
Correct Answer
x = 13, y = 12
Detailed Solution Steps
1
Step 1: For ∠CBD: The diagonal of a square bisects the 90° interior angle, so ∠CBD = 45°. Set up the equation: \(7x - 11 = 45\).
2
Step 2: Solve for x: Add 11 to both sides: \(7x = 56\). Divide by 7: \(x = 8\).
3
Step 3: For ∠AOD: The diagonals of a square are perpendicular, so ∠AOD = 90°. Set up the equation: \(5y + 30 = 90\).
4
Step 4: Solve for y: Subtract 30 from both sides: \(5y = 60\). Divide by 5: \(y = 12\).
Knowledge Points Involved
1
Square Diagonals Bisect Interior Angles
Each diagonal of a square splits a 90° vertex angle into two 45° angles, which is used to solve for x in this problem.
2
Perpendicular Diagonals of a Square
The diagonals of a square intersect at a 90° angle, which is used to set up the equation to solve for y.
3
Solving One-Variable Linear Equations
Linear equations are solved by isolating the variable using inverse operations (addition/subtraction followed by multiplication/division).
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