How to Find Side CD in a Triangle ABC with Given Properties: A Simple Guide

When dealing with triangles in geometry, understanding and applying trigonometric functions is key to solving many problems. In this guide, we will explore a specific problem where you need to find the length of side CD in a triangle ABC, given that AB 6 cm, the angle at A is 90°, and the angle at B is 30°. We will also discuss the use of trigonometric functions and some useful properties of right triangles.

The Properties and Given Information

We are given the following information about triangle ABC:

Since the sum of angles in a triangle is 180°, we can find the angle at C (angle C) as follows:

Angle C 180° - 90° - 30° 60°

Finding the Length of Altitude AD

Given that AD is an altitude, we can use the sine function to find its length:

Given that sine of 30° 0.5, we have:

AD AB * sin(30°) 6 * 0.5 3 cm

Calculating the Length of Side CD

To find the length of side CD, we can use the tangent function. First, we need to find the length of side AC. Since angle B 30°, we can use the cotangent function (which is the reciprocal of the tangent function):

x AD / tan(30°) 3 / (1/√3) 3√3 cm

Therefore, the length of AC is 3√3 cm. Now, to find the length of side CD, we use the tangent of angle B:

CD AC * tan(30°) 3√3 * (1/√3) 3 cm

This confirms that the length of CD is 3√3 cm.

Verifying the Solution Using the Altitude Formula

To verify our solution, we can use the altitude formula for right triangles:

For a right triangle with angle A 90°, the altitude can be found using the formula:

Altitude √(xy)

In this case, x AC and y DB. We already know AC 3√3 cm. To find DB, we use the fact that angle C 60°, so:

DB AB * cos(30°) 6 * (√3/2) 3√3 cm

Now, we can verify the altitude:

Altitude √(3√3 * 3√3) √(27) 3 cm

This confirms that our solution is correct.

Additional Information and Properties

Itrsquo;s worth noting that the use of circles can also be helpful in solving such problems. Any triangle with the hypotenuse as the diameter of a circle is a right triangle. If x 1, the altitude formula simplifies to:

Altitude √y

This can be a useful method for geometrically determining square roots.

Conclusion

With a good understanding of trigonometric functions like sine, cosine, and tangent, as well as the properties of right triangles, you can solve complex problems like finding side lengths quickly and accurately. The key is to break the problem down into smaller, manageable parts and apply the relevant trigonometric identities.