Solving for the Measure of ∠RAC in a Right Triangle ABC

Understanding the intricacies of right triangles can be a valuable skill in geometry, especially when dealing with specific angles and segments within the triangle. This article will explore how to solve for the measure of ∠RAC in triangle ABC, where

Understanding the Setup

Triangle ABC is a right triangle with ∠BAC 90°, where ∠B > ∠C. To begin our solution, we need to analyze the given properties of the triangle:

AP is an altitude from A to BC: This means that AP is perpendicular to BC. AQ is an angle bisector of ∠BAC: This means it divides ∠BAC into two equal parts. AR is a median of the triangle: AR extends from A to the midpoint R of BC. Given that ∠PAQ 13° and P is on BQ: This provides us with a specific angle and a point on a line segment.

Finding Angles

We start by noting that since AQ is the angle bisector of ∠BAC, it divides the angle into two equal parts:

∠BAQ ∠CAQ 1/2 × 90° 45°

Given that ∠PAQ 13°, we can find ∠BAP as follows:

∠BAP ∠BAQ - ∠PAQ 45° - 13° 32°

AP is also an altitude, which means that triangle BPA is a right triangle, making ∠B 58° because the sum of angles in a triangle is 180° and ∠AMP (90°) is given by:

∠B 180° - 90° - 32° 58°

Thus, ∠C 180° - 90° - 58° 32°

Conclusion

Since AR is a median, R is the midpoint of BC. In a right triangle, the midpoint of the hypotenuse is the circumcenter, making AR and CR the circumradii. As triangle ARC is isosceles, the base angles ∠RAC and ∠CAR are equal:

∠RAC ∠CAR 32°

Therefore, the measure of ∠RAC is boxed{32°}.

Key Takeaways

Right Triangle Properties: The properties of a right triangle, especially those involving the altitude, angle bisector, and median, can be used to deduce specific angle measures. Angle Bisectors: An angle bisector divides an angle into two equal parts. Medians in Right Triangles: The median to the hypotenuse in a right triangle forms an isosceles triangle with the circumcenter.

Frequently Asked Questions (FAQ)

Q: What is the purpose of using angle bisectors in a right triangle?

A: Angle bisectors in a right triangle help to divide angles into equal parts, which can be useful in solving for other angles within the triangle.

Q: How does the median of a right triangle help in solving for angles?

A: The median to the hypotenuse in a right triangle is also the circumradius and creates an isosceles triangle with the midpoint of the hypotenuse as the circumcenter.

Q: Can the altitude of a right triangle be used to find specific angles?

A: Yes, the altitude in a right triangle is perpendicular to the base, and this perpendicularity can be used to determine other angles within the triangle.

By understanding these properties and using them effectively, one can solve complex problems involving right triangles.