let ABC be a right angled triangle with ab 6 CM BC 8 cm let D be a point on the hypotenuse AC such that CBD equal to 60 degree what is the length in CM of BD

We are given a right-angled triangle \( \triangle ABC \), where:

- \( AB = 6 \, \text{cm} \)
- \( BC = 8 \, \text{cm} \)
- \( \angle ABC = 90^\circ \)
- \( D \) is a point on the hypotenuse \( AC \), such that \( \angle CBD = 60^\circ \).

We are tasked with finding the length of \( BD \).

### Step 1: Find the length of the hypotenuse \( AC \)

Using the Pythagorean theorem for \( \triangle ABC \):

\[
AC^2 = AB^2 + BC^2
\]
\[
AC^2 = 6^2 + 8^2 = 36 + 64 = 100
\]
\[
AC = \sqrt{100} = 10 \, \text{cm}
\]

### Step 2: Use the Law of Sines in \( \triangle BCD \)

We know the following angles and lengths:
- \( \angle CBD = 60^\circ \)
- \( \angle ABC = 90^\circ \), so \( \angle ACB = 30^\circ \) (since angles in a triangle sum to 180°).
- The length \( BC = 8 \, \text{cm} \).

Now, applying the Law of Sines in \( \triangle BCD \):

\[
\frac{BD}{\sin \angle ACB} = \frac{BC}{\sin \angle CBD}
\]
\[
\frac{BD}{\sin 30^\circ} = \frac{8}{\sin 60^\circ}
\]
\[
\frac{BD}{0.5} = \frac{8}{\frac{\sqrt{3}}{2}} = \frac{8 \times 2}{\sqrt{3}} = \frac{16}{\sqrt{3}}
\]
\[
BD = \frac{16}{\sqrt{3}} \times 0.5 = \frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3} \, \text{cm}
\]

Thus, the length of \( BD \) is \( \frac{8\sqrt{3}}{3} \) cm, approximately \( 4.62 \, \text{cm} \).

---

Would you like me to explain any part in more detail or explore related topics?

Here are some related questions you might find useful:
1. How does the Law of Sines apply to non-right triangles?
2. How can we calculate the area of a triangle using trigonometry?
3. What are the conditions to use the Law of Cosines in a triangle?
4. How can we derive the relationship between angles in any triangle?
5. What is the geometric interpretation of the Pythagorean theorem?

**Tip**: The Law of Sines is especially useful when dealing with non-right triangles, allowing you to find unknown lengths and angles.

Let ABC be a right-angled triangle with AB = 6 cm, BC = 8 cm. Let D be a point on the hypotenuse AC such that ∠CBD = 60°. What is the length of BD in cm?

Solve the problem involving a right-angled triangle ABC with sides AB = 6 cm and BC = 8 cm, and a point D on the hypotenuse AC such that ∠CBD = 60°. Learn how to apply the Pythagorean theorem and the Law of Sines to find the length of BD. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation