Problem 17 · 2012 Math Kangaroo
Stretch
Geometry & Measurement
area
A right-angled triangle with side lengths a = 8, b = 15 and c = 17 is given. How big is the radius r of the inscribed semicircle shown?
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Answer: D — 4·8
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Hint 1 of 2
The semicircle's flat edge lies on one leg, so its centre sits on that leg a distance \(r\) from the right-angle corner.
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Hint 2 of 2
The curved part is tangent to the hypotenuse, so the centre is also a distance \(r\) from the hypotenuse line.
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Approach: equal distances from centre to the two tangent sides
- Put the right angle at the origin with the legs along the axes; the hypotenuse is the line \(8x + 15y = 120\).
- The centre sits at \((r,0)\); its distance to the hypotenuse is \(\frac{120 - 8r}{17}\) and must equal \(r\).
- Then \(120 - 8r = 17r\) gives \(25r = 120\), so \(r = 4.8\), choice D.
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