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We are tasked with finding the length of \( BD \), given the right-angled triangle \( \triangle ABC \) with a right angle at \( B \), where:

- \( AB = 8 \, \text{cm} \)
- \( BC = 6 \, \text{cm} \)
- \( BD \perp AC \) (i.e., \( BD \) is perpendicular to \( AC \), and \( D \) is a point on \( AC \)).

To approach this problem, we can first calculate the hypotenuse \( AC \) using the Pythagorean theorem, as \( \triangle ABC \) is a right triangle.

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

Now, knowing that \( BD \perp AC \), \( BD \) is the height of the triangle from vertex \( B \) to the hypotenuse \( AC \). We can calculate the area of \( \triangle ABC \) in two ways:

1. Using the base \( AB \) and height \( BC \):

\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 8 \times 6 = 24 \, \text{cm}^2
\]

2. Using the hypotenuse \( AC \) and height \( BD \):

\[
\text{Area of } \triangle ABC = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 10 \times BD
\]

Equating the two expressions for the area:

\[
24 = \frac{1}{2} \times 10 \times BD
\]

Solving for \( BD \):

\[
BD = \frac{24 \times 2}{10} = \frac{48}{10} = 4.8 \, \text{cm}
\]

Thus, the length of \( BD \) is \( 4.8 \, \text{cm} \), which corresponds to option B.

Let me know if you'd like any further details or have any questions! Here are some related questions to think about:

1. What is the area of triangle \( \triangle ABC \) if \( AB = 10 \, \text{cm} \) and \( BC = 12 \, \text{cm} \)?
2. How do you find the length of the height in an isosceles triangle if the base and one side are known?
3. Can you calculate the hypotenuse for any right triangle with sides \( a = 7 \, \text{cm} \) and \( b = 24 \, \text{cm} \)?
4. How do you verify if a triangle is a right triangle using only the side lengths?
5. What are the different ways to calculate the area of a triangle based on its given sides?

**Tip:** In problems involving right triangles, the Pythagorean theorem is often useful for calculating unknown side lengths!

Perhatikan gambar ∆ABC dibawah ini! Segitiga tersebut siku-siku di B dengan AB = 8 cm dan BC = 6 cm. Titik D terletak di sisi AC sedemikian sehingga BD ⊥ AC. Panjang BD adalah…

This problem involves finding the length of BD in a right-angled triangle ∆ABC where AB = 8 cm, BC = 6 cm, and BD ⊥ AC. Using the Pythagorean theorem and area calculations, discover how to determine the correct length of BD. Suitable for students in Grades 8-10.

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