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The triangle STU has sides of length 4, 7, and x - Junior Cycle Mathematics - Question b - 2015

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The triangle STU has sides of length 4, 7, and x. Find the two values of x for which △ STU is a right-angled triangle. Give each answer in surd form.

Worked Solution & Example Answer:The triangle STU has sides of length 4, 7, and x - Junior Cycle Mathematics - Question b - 2015

Step 1

Find the first value of x for which △ STU is a right-angled triangle

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Answer

Using the Pythagorean theorem for the triangle where the side lengths are 4 and 7, we set up the equation: 42+x2=724^2 + x^2 = 7^2 Solving this, we have: 16+x2=4916 + x^2 = 49 Thus, we get: x2=49−16x^2 = 49 - 16 x2=33x^2 = 33 Taking the square root: x=ext√33x = ext{√33}

Step 2

Find the second value of x for which △ STU is a right-angled triangle

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Answer

Now, we consider the scenario where 7 is one of the shorter sides, leading us to: 42+72=x24^2 + 7^2 = x^2 Substituting the values, we get: 16+49=x216 + 49 = x^2 This simplifies to: 65=x265 = x^2 Taking the square root yields: x=ext√65x = ext{√65}

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