Question

In ΔABC, M, N, P are midpoints of sides AB, AC and BC. X, Y, Z are midpoints of sides of ΔMNP. If XY = 2 cm, YZ = 2.5 cm, XZ = 3 cm, then sides of ΔABC are ______.

Options

• 4 cm, 5 cm, 6 cm

• 8 cm, 10 cm, 12 cm

• 6 cm, 7.5 cm, 9 cm

• none of these

Answer 1

In ΔABC, M, N, P are midpoints of sides AB, AC and BC. X, Y, Z are midpoints of sides of ΔMNP. If XY = 2 cm, YZ = 2.5 cm, XZ = 3 cm, then sides of ΔABC are 8 cm, 10 cm, 12 cm.

Explanation:

Given:

• M, N, and P are midpoints of sides AB, AC and BC of ΔABC, 
• X, Y and Z are midpoints of the sides of ΔMNP,
• Lengths of sides of ΔXYZ are:
  • XY = 2 cm
  • YZ = 2.5 cm
  • XZ = 3 cm

Step 1: Understand the relationships between the triangles

• ΔMNP is formed by joining midpoints of ΔABC. So, each side of ΔMNP is half the length of the corresponding side of ΔABC.
• Similarly, ΔXYZ is formed by joining midpoints of ΔMNP, so each side of ΔXYZ is half the length of the corresponding side of ΔMNP.

Step 2: Calculate the sides of ΔMNP

Since `XY = 1/2 MN`, we have:

MN = 2 × XY = 2 × 2 = 4 cm

Similarly,

NP = 2 × YZ = 2 × 2.5 = 5 cm

PM = 2 × XZ = 2 × 3 = 6 cm

Step 3: Calculate the sides of ΔABC

Each side of ΔMNP is half the length of the corresponding side of ΔABC, so:

AB = 2 × MN = 2 × 4 = 8 cm

BC = 2 × NP = 2 × 5 = 10 cm

CA = 2 × PM = 2 × 6 = 12 cm