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Calculate Side and Diagonal Lengths of a Square with Side 15
Mathematics
Grade 8 (Junior High School)
Question Content
In square FUTV with side length 15, find the lengths of VU, SU, TV, and SW, where S is the intersection of the diagonals.
Correct Answer
VU = 15, SU = 21.21, TV = 21.21, SW = 10.6
Detailed Solution Steps
1
Step 1: Identify VU as a side of the square, so VU equals the given side length: VU = 15.
2
Step 2: Calculate the length of the square's diagonal using the formula \(d = s\sqrt{2}\) (where \(s\) is side length). Substitute \(s=15\): \(d = 15\sqrt{2} \approx 21.21\). TV is a diagonal, so TV = 21.21, and SU is also a diagonal, so SU = 21.21.
3
Step 3: The diagonals of a square bisect each other, so SW is half the length of diagonal SU. Calculate \(SW = \frac{21.21}{2} = 10.6\).
Knowledge Points Involved
1
Properties of Square Sides
All sides of a square are congruent, meaning they have equal length. This is a fundamental property used to identify side lengths directly from a given side measurement.
2
Square Diagonal Length Formula
The diagonal \(d\) of a square with side length \(s\) is calculated as \(d = s\sqrt{2}\). This comes from applying the Pythagorean theorem to the right triangle formed by two sides and a diagonal of the square.
3
Diagonals of a Square Bisect Each Other
The two diagonals of a square intersect at their midpoints, so each segment of a diagonal from the intersection to a vertex is half the total length of the diagonal.
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