Using the given information, length of QR = 16.6 cm
The area of the trapezium is 304.5 cm²
<h3>Calculating area of a trapezium</h3>
From the question we are to calculate the length of QR and the area of trapezium PQRS
In the given diagram,
Let the midpoint of PS be T
Then, we can write that
|OP|² = |OT|² + |PT|² (<em>Pythagorean theorem</em>)
|OP| = radius = 13 cm
|PT| = 1/2 × PS = 1/2 × 24 cm = 12 cm
∴ 13² = |OT|² + 12²
169 = |OT|² + 144
|OT|² = 169 -144
|OT|² = 25
|OT| = √25
|OT| = 5 cm
Also, let the midpoint of QR be U
Then, we can write that
|OQ|² = |OU|² + |QU|² (<em>Pythagorean theorem</em>)
|OQ| = radius = 13 cm
From the given information,
The distance of QR from O is twice the distance of PS from O
∴ |OU| = 2 × |OT|= 2 × 5 cm = 10cm
Thus,
13² = ||² + 12²
169 = 10² + |QU|²
|QU|² = 169 -100
|QU|² = 69
|QU| = √69
|QU| = 8.3 cm
Now,
Length of QR = 2 × |QU| = 2 × 8.3 cm
Length of QR = 16.6 cm
b)
Area of the trapezium = 1/2(|QR| + |PS|) × |TU|
Area of the trapezium = 1/2(16.6 + 24) × 15
NOTE: |TU| = |OT| + |OU|
Area of the trapezium = 1/2(40.6) × 15
Area of the trapezium = 20.3 × 15
Area of the trapezium = 304.5 cm²
Hence, the area of the trapezium is 304.5 cm²
Learn more on Calculating area of trapezium here: brainly.com/question/3435635
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