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Let's solve each part of the problem based on the information provided:

### Given Data:
- \( AB = 6 \, \text{cm} \)
- \( BC = 8 \, \text{cm} \)
- \( L \) is the midpoint of \( AC \).
- \( M \) is the midpoint of \( BC \).
- \( BL = 5 \, \text{cm} \).

### Part (a): Calculate \( LM \).

Since \( L \) and \( M \) are midpoints of \( AC \) and \( BC \) respectively, we can apply the **Midline Theorem** (or Midsegment Theorem), which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Thus, \( LM \) will be:
\[
LM = \frac{1}{2} \times AB = \frac{1}{2} \times 6 = 3 \, \text{cm}
\]

So,
\[
LM = 3 \, \text{cm}
\]

### Part (b): Show that \( \triangle BLM \) is a right triangle at \( M \).

To prove that \( \triangle BLM \) is a right triangle at \( M \), we can use the **Pythagorean Theorem**. We already know:
- \( BL = 5 \, \text{cm} \)
- \( LM = 3 \, \text{cm} \)

Now, let's find \( BM \):
Since \( M \) is the midpoint of \( BC \), \( BM \) will be half of \( BC \):
\[
BM = \frac{1}{2} \times BC = \frac{1}{2} \times 8 = 4 \, \text{cm}
\]

Now, check if \( BL^2 = BM^2 + LM^2 \):
\[
BL^2 = 5^2 = 25
\]
\[
BM^2 + LM^2 = 4^2 + 3^2 = 16 + 9 = 25
\]

Since \( BL^2 = BM^2 + LM^2 \), by the Pythagorean Theorem, \( \triangle BLM \) is a right triangle at \( M \).

### Part (c): Determine the nature of \( \triangle ABC \).

We can determine the nature of \( \triangle ABC \) by comparing the lengths \( AB \), \( BC \), and \( AC \).

Using the Pythagorean Theorem to find \( AC \):
\[
AC = 2 \times AL \quad \text{(since \( L \) is the midpoint of \( AC \))}
\]

Since \( BL = 5 \) cm and \( AB = 6 \) cm, we can use the fact that \( \triangle BLM \) is right-angled at \( M \) to solve for \( AC \) as the hypotenuse in \( \triangle ABC \).

This

Given AB = 6 cm, BC = 8 cm, L is the midpoint of [AC] and M is the midpoint of [BC] with BL = 5 cm. Calculate LM, show that BLM is a right triangle at M, and determine the nature of triangle ABC.

Solve a geometry problem involving triangle ABC where AB = 6 cm, BC = 8 cm, and midpoints L and M. Calculate LM, prove BLM is a right triangle, and analyze the type of triangle ABC. Suitable for students in Grades 8-10.

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