In a right-angled triangle ABC, angle A = 90 degrees, AB = 13 cm, height AD = 12 cm. Find AC and CosC?

univerkov | January 12, 2021 | education

Since BP is the height of the triangle, the AВD and AСD triangles are rectangular.
Then in a right-angled triangle AВD, ВD^2 = AB ^ 2 – AD ^ 2 = 169 – 144 = 25.
ВD = 5 cm.
Rectangular triangles AВD and AСD are similar in acute angle.
Then AC /AD = AВ / ВD.
AC = AD * AB / ВD = 12 * 13/5 = 31.2 cm.
In a right-angled triangle of AСD, according to the Pythagorean theorem, we determine the length of the leg of the СD.
СD ^ 2 = AC ^ 2 – AD ^ 2 = 973.44 – 144 = 829.44.
СD = 28.8 cm.
Determine the cosine of the angle C.
CosC = CD / AC = 28.8 / 31.2 = 12/13.
Answer: The length of the AC leg is 31.2 cm, CosC = 12/13.

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