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Question

Difficulty: EasyProperties of Quadrilaterals

In rectangle PQRSPQRS, the length of side PQPQ is 8 inches and the length of diagonal PRPR is 10 inches. What is the length, in inches, of side QRQR?

  1. A
    2
  2. 6Answer
  3. C
    6.4
  4. D
    12.8
  5. E
    18

Answer

The length of side QR is 6 inches.
The correct answer is 6. In a rectangle, all four interior angles are right angles (90 degrees). Therefore, triangle PQR is a right triangle with the right angle at Q. According to the Pythagorean theorem, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: PQ² + QR² = PR². Substituting the given values: 8² + QR² = 10², which simplifies to 64 + QR² = 100. Subtracting 64 from both sides gives QR² = 36. Taking the square root of both sides yields QR = 6 inches.

Step-by-Step Solution

1
Identify the geometric relationship in rectangle PQRS.
Angle PQR is a right angle (90 degrees), which makes triangle PQR a right triangle with legs PQ and QR, and hypotenuse PR.
By definition, all interior angles of a rectangle are 90 degrees, and the diagonal connects opposite vertices, forming two right triangles.
2
Apply the Pythagorean theorem to triangle PQR.
PQ² + QR² = PR², which becomes 8² + QR² = 10².
The Pythagorean theorem states that in any right triangle, the sum of the squares of the leg lengths equals the square of the hypotenuse length.
3
Solve for the unknown side length QR.
64 + QR² = 100, which gives QR² = 36, and thus QR = 6.
Subtract 64 from both sides to isolate QR², then take the square root of 36.

Key Concept

Properties of rectangles and the application of the Pythagorean theorem to right triangles formed by diagonals.
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Topics

Properties of Quadrilaterals

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