Find the acute angles of a right-angled triangle ABC, if the height BD, drawn to the hypotenuse

| August 4, 2021 | education

Find the acute angles of a right-angled triangle ABC, if the height BD, drawn to the hypotenuse, cuts off a segment DC equal to 9 cm from the hypotenuse AC; BC = 6√3cm.

Consider the resulting triangles ABC and BDC in order to consider the acute angle <BCA in them. Consider the angle function:

cоs <C = DC / ВС. Based on the data:

cоs <C = 9 / (6 * √3) = 3/2 * √3 = (3 * √3) / (2 * √3√3) = (3 * √3) / 2 * 3 = √3 / 2.

We look at the tabular data, the value of the cosine of the angle, equal to √3 / 2, corresponds to the angle <C = 30 °. Then the angle <A = 90 ° – 30 ° = 60 °.

To find out <C from triangle ABC, you need to determine AC from the similarity formula of triangles ABC and BCD:

AC: BC = BC: DC. AC = BC² / DC = (6 * √3) ² / 9 = 36 * 3/9 = 12,

cos <C = BC / AC = 6 * √3 / 12 = √3 / 2. <C = 30 °.



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