In the following triangle,
AB ⊥ CD and AC ⊥ BC
CD is the bisector of triangle ABC
If AD = 16m, and CD = 8m, find BD
To find BD in the given triangle, we can use the concept of angle bisectors in a triangle. Here's how you can approach the problem:
1. First, draw the triangle ABC with the given information. Label the points as A, B, C, D accordingly.
2. Since AB ⊥ CD, it means that AB is perpendicular to CD. This implies that angle ABD is a right angle (90 degrees).
3. Since AC ⊥ BC, it means that AC is perpendicular to BC. This implies that angle BAC is also a right angle (90 degrees).
4. Now, observe that CD is the bisector of triangle ABC. This means that it divides angle ABC into two equal angles. Let's call these angles x.
5. Note that angle ABD + angle CDB = 90 degrees (sum of angles in a triangle). Therefore, angle ADB = 90 - x.
6. In triangle ADB, you have a right-angled triangle with AD = 16m and angle ADB = 90 - x. You can use trigonometry (specifically, the cosine function) to find the length of BD.
7. We know that cos(90 - x) = BD / AD. Rearranging this equation, we get BD = AD * cos(90 - x).
8. Substituting the given values, BD = 16m * cos(90 - x).
Now, you can calculate the value of BD using a calculator or by using the trigonometric table for the cosine function.