Question

In ΔABC, M, N and P are mid-points of sides AB, AC and BC respectively. X, Y and Z are mid-points of sides of ΔMNP. If XY = 2.5 cm, YZ = 3.5 cm and XZ = 4 cm, find the sides of ΔABC.

Answer 1

We are given the following information:

• M, N and P are midpoints of the sides of ΔABC,
• X, Y and Z are midpoints of the sides of ΔMNP,
• The lengths of the sides of ΔXYZ are:
  • XY = 2.5 cm,
  • YZ = 3.5 cm,
  • XZ = 4 cm.

We are tasked with finding the sides of ΔABC.

Step 1: Use of Midpoint Theorem

From the Midpoint Theorem, we know that when we join the midpoints of the sides of a triangle, the resulting triangle is similar to the original triangle and its sides are half the length of the corresponding sides of the original triangle.

1. In ΔMNP, the midpoints of the sides of ΔABC, the sides of ΔMNP are half the lengths of the sides of ΔABC.

So, the sides of ΔMNP are:

• `MN = 1/2 AB`,
• `NP = 1/2 BC`,
• `MP = 1/2 AC`.

2. Similarly, in ΔXYZ, the midpoints of the sides of ΔMNP, the sides of ΔXYZ are half the lengths of the sides of ΔMNP.

Therefore:

• `XY = 1/2 MN`,
• `YZ = 1/2 NP`,
• `XZ = 1/2 MP`.

Step 2: Relate the sides of ΔXYZ to the sides of ΔABC

We are given the lengths of the sides of ΔXYZ:

• XY = 2.5 cm,
• YZ = 3.5 cm,
• XZ = 4 cm.

Using the relationships from the Midpoint Theorem:

`XY = 1/2 xx MN = 2.5  cm`

⇒ MN = 2 × 2.5 = 5 cm

`YZ = 1/2 xx NP = 3.5  cm`

⇒ NP = 2 × 3.5 = 7 cm

`XZ = 1/2 xx MP = 4  cm`

⇒ MP = 2 × 4 = 8 cm

Step 3: Calculate the sides of ΔABC

From the relationships between the sides of ΔMNP and ΔABC:

`MN = 1/2 AB`

⇒ AB = 2 × MN

= 2 × 5 

AB = 10 cm

`NP = 1/2 BC`

⇒ BC = 2 × NP

= 2 × 7 

BC = 14 cm

`MP = 1/2 AC`

⇒ AC = 2 × MP

= 2 × 8 

AC = 16 cm