Question

In a right triangle ABC, line CH is the altitude to the hypotenuse. find the ratio of AH:HB if measure of angle a = to 30 degrees.

Answer 1

In a right triangle \( ABC \) where \( \angle C \) is the right angle, and \( CH \) is the altitude to the hypotenuse \( AB \), we want to find the ratio \( AH:HB \) when \( \angle A = 30^\circ \).

First, we can use some properties of right triangles and the relationships between the angles.

1. Since \( \angle A = 30^\circ \) and \( \angle C = 90^\circ \), it follows that:
\[
\angle B = 90^\circ - \angle A = 90^\circ - 30^\circ = 60^\circ
\]

2. In a right triangle, the altitude to the hypotenuse creates two smaller triangles, \( \triangle AHC \) and \( \triangle BHC \), both of which are also right triangles.

3. From the properties of a 30-60-90 triangle, we know the lengths of the sides. If we let the length of side \( BC \) (which is opposite to \( \angle A \)) be \( a \), then:
- \( AC = a \sqrt{3} \) (the side opposite \( \angle B \)),
- The hypotenuse \( AB = 2a \).

4. The ratio of the segments \( AH \) and \( HB \) can be found using the relationship that states:
\[
\frac{AH}{HB} = \frac{AC^2}{BC^2}
\]

5. Substituting the lengths from our triangle:
- Since \( AC = a\sqrt{3} \) and \( BC = a \):
\[
\frac{AH}{HB} = \frac{(a\sqrt{3})^2}{a^2} = \frac{3a^2}{a^2} = \frac{3}{1}.
\]

Thus, the ratio \( AH:HB \) is:
\[
\boxed{3:1}.
\]