Question

M and N are midpoints of sides AB and BC of ΔABC. If AC = 12 cm and ∠BMN = 55°. Then

1. MN = 6 cm
2. MN || AC
3. ∠c = 55°

Which statements from the above are valid?

Options

• only 1

• only 2

• 1 and 2

• 2 and 3

Answer 1

1 and 2

Explanation:

Given:

• M and N are the midpoints of sides AB and BC of  triangle ΔABC.
• AC = 12 cm
• ∠BMN = 55°

We are asked to determine which of the following statements are valid:

1. MN = 6 cm
2. MN || AC
3. ∠C = 55°

Step 1: Analyze Statement 1: MN = 6 cm

By the Midpoint Theorem, in triangle ΔABC, the segment MN joining the midpoints of AB and BC is parallel to side AC and is half the length of AC.

Given that AC = 12 cm, we have:

`MN = 1/2 xx AC`

= `1/2 xx 12`

= 6 cm

Thus, Statement 1 is valid.

Step 2: Analyze Statement 2: MN || AC

Again, by the Midpoint Theorem, since M and N are the midpoints of sides AB and BC respectively, the segment MN is parallel to side AC.

Therefore, Statement 2 is valid.

Step 3: Analyze Statement 3: ∠C = 55°

We are given that ∠BMN = 55°, but this is the angle between the segment MN and side BC. There is no direct relationship between ∠BMN and ∠C without additional information.

Hence, Statement 3 is not necessarily valid.