Question

In a right triangle, the bisector of the right angle divides the hypotenuse in the ratio of 3 is to 5. Determine the measures of the included angles of the triangle.

Answer 1

using angle bisector theorem:
\[\frac{ 3x }{ 5x }=\frac{ a }{ b }\]
so \[a=\frac{ 3b }{ 5 }\]
\[\tan \theta =\frac{ a }{ b }=\frac{ 3b }{5b }=\frac{ 3 }{ 5} \rightarrow \theta =31^o\]
\[\alpha = 180^o-31^o-90^o=59^o\]

Answer 2

A right triangle with a 5 unit long hypotenuse will have the opposite leg of 3 units and the adjacent leg of 4 units. There are only two numbers which make a right triangle when having a 5 unit hypotenuse. These are 3 and 4, with the 3 unit leg being the vertical leg of the right triangle.

Based on this, the sine of the smaller angle will be 3 / 5 which is 0.6
.
The angle having a sine of 0.6 is 36.87 degrees. The other acute angle will be 90 - 36.87 = 53.13 degrees.