In an acute-angled triangle ABC, the heights AK and CE are drawn, CE = 12cm, BE = 9 cm, AK = 10 cm. Find AC.
| January 28, 2021 | education
From the right-angled triangle CBE, by the Pythagorean theorem, we determine the length of the hypotenuse BC.
BC ^ 2 = CE ^ 2 + BE ^ 2 = 144 + 81 = 225.
BC = 15 cm.
By the property of the heights of the triangle, the heights of the triangle are inversely proportional to the sides to which they are drawn.
Then AK / CE = (1 / CD) / (1 / AB).
(1 / AB) = (CE / CB) / AK.
AB = AK / (CE / CD) = 10 / (12/15) = 12.5 cm.
Then the length of the segment AE = AB – BE = 12.5 – 9 = 3.5 cm.
From the right-angled triangle ACE, we determine the length of the hypotenuse AB.
AB ^ 2 = CE ^ 2 + AE ^ 2 = 144 + 12.25 = 156.25.
AB = 12.5 m.
Answer: The length of the AB side is 12.5 cm.
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