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The triangle PQR has sides of length 8, 11, and y - Junior Cycle Mathematics - Question 12 - 2015
Question 12
The triangle PQR has sides of length 8, 11, and y. Write down one value of y for which \( \angle PQR \) is an isosceles triangle. \( y = \) The triangle STU has si... show full transcript
Worked Solution & Example Answer:The triangle PQR has sides of length 8, 11, and y - Junior Cycle Mathematics - Question 12 - 2015
Step 1
Write down one value of y for which \( \angle PQR \) is an isosceles triangle.
Answer
To determine the value of y that makes triangle PQR isosceles, we need two sides to be equal. Hence, we have two scenarios:
- If y = 8, then the triangle has sides 8, 8, and 11.
- If y = 11, then the triangle has sides 11, 11, and 8. Therefore, one possible value of y is ( y = 8 ) or ( y = 11 ).
Step 2
Find the two values of x for which \( \angle STU \) is a right-angled triangle.
Answer
To find the values of x for which triangle STU is right-angled, we can use the Pythagorean theorem:
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For the right angle opposite to side 7: [ 4^2 + x^2 = 7^2 ] [ 16 + x^2 = 49 ] [ x^2 = 33 ] [ x = \sqrt{33} ]
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For the right angle opposite to side 4: [ 4^2 + 7^2 = x^2 ] [ 16 + 49 = x^2 ] [ x^2 = 65 ] [ x = \sqrt{65} ]
Thus, the two values of x are ( x = \sqrt{33} ) and ( x = \sqrt{65} ).