Solving Right Triangle Problems: Application of Properties of 30-60-90 Triangles

Introduction: In this article, we delve into a classic problem involving a right triangle, its altitude, and an angle bisector. The focus will be on demonstrating the application of the properties of 30-60-90 triangles and how they aid in solving complex geometric problems. This content is designed to be search engine optimized (SEO) for the terms 'Right Triangle', '30-60-90 Triangle', and 'Altitude'.

In the right triangle ABC, with C being the right angle, the altitude CH to the hypotenuse AB intersects the angle bisector AL at point D. Given AD 8 and DH 4, we aim to determine the length of BC using the properties of 30-60-90 triangles.

Problem Statement and Solution

Given the right-angled triangle ABC, where C is the right angle, and the altitude CH to the hypotenuse AB intersects the angle bisector AL at point D. It is also given that AD 8 and DH 4. We need to determine the length of BC.

Let's start by recognizing that triangle ADH is a 30-60-90 triangle due to the given length ratios. The properties of a 30-60-90 triangle can be summarized as follows:

The length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is the product of the shorter leg and the square root of 3.

Using the fact that triangle ADH is a 30-60-90 triangle, we can determine the length of AH and CH. Since AD is the hypotenuse, we have:

AD 8

Using the property of the hypotenuse, we find:

AH AD * Tan(30°) 8 * (1/√3) 8 * (1/1.732) ≈ 4.62

Since angle DAH 30°, we can use the Pythagorean theorem to find AC:

AC AH / Sin(30°) 4.62 / 0.5 8.34

Now, using the properties of the 30-60-90 triangle, we find BC:

BC AC * √3 ≈ 8.34 * 1.732 ≈ 14.48

However, this method seems to deviate from the given solution. Let's reconsider the problem with a simpler approach.

Revised Approach

Given AD 8 and DH 4, since triangle ADH is a 30-60-90 triangle, we have:

AD 8 (hypotenuse) DH 4 (shorter leg) AH √(AD^2 - DH^2) √(8^2 - 4^2) √(64 - 16) √48 4√3 ≈ 6.93

Since triangle ADH is a 30-60-90 triangle, and AH is the long leg, we can infer that A 60°. Consequently, triangle ACH is a 30-60-90 triangle as well, with DCH 30°. We can now apply the properties of 30-60-90 triangles to find the lengths of AC and BC:

AH 4√3 ≈ 6.93 C 90°, so AC AH * 2 4√3 * 2 8√3 ≈ 13.86 (hypotenuse) BC AC * Tan(60°) 8√3 * √3 24 (since Tan(60°) √3)

Final Answer

Therefore, the length of BC is 24 units.

Additional Information

For a right-angled triangle ABC with C 90°, CH is the altitude from C to the hypotenuse AB. AL is the angle bisector of angle A intersecting CH at D. Given AD 8 and DH 4, we can determine the following:

ADH is a 30-60-90 triangle, so angle DAH 30° and A 60°. AH √(AD^2 - DH^2) √(64 - 16) √48 4√3 (approximately 6.93). AC AH * 2 8√3 (approximately 13.86). BC AC * Tan(60°) 24.

The length of AB, the hypotenuse, can be calculated as:

AB 2 * AC 2 * 8√3 ≈ 16.97 units.