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Solve for Side, Diagonal Lengths and Angles in a Square with Given Segment Lengths
Mathematics
Grade 8 (Junior High School)
Question Content
In square WZYX with diagonals intersecting at R, given WR = 18 and XY = 26, find: a) ZY, b) WY, c) RX, d) m∠WRZ, e) m∠XYZ, f) m∠ZWT.
Correct Answer
a) ZY = 26, b) WY = 36, c) RX = 18, d) m∠WRZ = 90°, e) m∠XYZ = 90°, f) m∠ZWT = 45°
Detailed Solution Steps
1
Step 1: All sides of a square are congruent, so ZY = XY = 26.
2
Step 2: Diagonals of a square bisect each other, so diagonal WY = 2×WR = 2×18 = 36.
3
Step 3: Diagonals of a square are congruent and bisect each other, so RX = WR = 18.
4
Step 4: Diagonals of a square are perpendicular, so m∠WRZ = 90°.
5
Step 5: All interior angles of a square are right angles, so m∠XYZ = 90°.
6
Step 6: The diagonal bisects the interior right angle, so m∠ZWT = \(\frac{90°}{2} = 45°\).
Knowledge Points Involved
1
Congruent Sides of a Square
Every side of a square has the same length, so any side can be determined if one side length is known.
2
Congruent and Bisecting Diagonals of a Square
The two diagonals of a square are equal in length, and they intersect at their midpoints, so each segment from the intersection to a vertex is half the diagonal length.
3
Perpendicular Diagonals of a Square
The diagonals of a square intersect at a right angle (90°), forming four right triangles within the square.
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