Introduction

Understanding the different measurements in a triangle, such as the length of AD (altitude, median, or angle bisector), is crucial for optimizing triangle-related problems. This article explores how to calculate the length of AD in various scenarios, providing a comprehensive guide that can help students and professionals alike enhance their problem-solving skills.

Given Lengths: AB 12 cm, BC 15 cm, CA 9 cm

1. Perpendicular AD

When dealing with a right-angled triangle where AB 12 cm, BC 15 cm, and CA 9 cm, we can calculate the length of the altitude AD from point A to the side BC using the area of the triangle.

Solution

First, we verify if the triangle is right-angled at A by using the Pythagorean theorem:

BA2 CA2 12 cm2 9 cm2 144 cm2 81 cm2 225 cm2

BC2 15 cm2 225 cm2

Since BA2 CA2 BC2, the triangle is indeed right-angled at A.

The area of triangle ABC can be calculated in two ways:

Solving for AD:

AD × BC / 2 AB × AC / 2

AD × 15 cm 12 cm × 9 cm

AD (12 cm × 9 cm) / 15 cm 36 / 5 cm 7.2 cm

The length of AD is 7.2 cm.

2. Median AD

When AD is a median, it is a line segment joining a vertex to the midpoint of the opposite side.

Solution

The triangle is right-angled at A, and D is the midpoint of BC:

BD CD BC / 2 15 cm / 2 7.5 cm

Since D is the midpoint of the hypotenuse, AD BD CD 7.5 cm.

The length of AD is 7.5 cm.

3. Angle Bisector AD

In a right-angled triangle, if AD is the angle bisector of angle BAC, it divides the triangle into two smaller triangles.

Solution

Since BAC 90°, BAD CAD 45°. Using the area relationship:

Area of triangle BAD (1/2) × AB × AD × sin(45°)

Area of triangle CAD (1/2) × AC × AD × sin(45°)

(1/2) × 12 cm × AD × sin(45°) / (1/2) × 9 cm × AD × sin(45°) (12 cm × 9 cm) / (21 sin(45°) × AD)

12 cm × AD × 21/2 / (2 × 7) 12 cm × 9 cm / (21 sin(45°) × AD)

AD (36 × 21/2) / (14) ≈ 7.272 cm

The length of AD is approximately 7.272 cm.

These calculations demonstrate the versatility of using different properties of triangles (right-angled, median, angle bisector) to find the length of AD. Understanding these concepts is essential for solving complex geometric problems and optimizing your approach using search engine optimization (SEO).