Question

In ΔABC, M and N are mid-points of sides AB and AC respectively.

1. If MN = 4x – 2 and BC = 6x + 3, find x.
2. If ∠MNC = (3y + 10)° and ∠C = (y + 10)°, find y. 
3. If ∠AMN = (2a + 15)° and ∠B = (3a – 15)°, find a. 
4. If AB = 7 cm, BC = 10 cm, AC = 9.2 cm, find the perimeter of MNCB.

Answer 1

Step 1:

`MN = 1/2 BC`

`4x - 2 = 1/2(6x + 3)`

`8x - 4 = 6x + 3`

`2x = 7`

`x = 7/2 = 3.5`

Step 2:

Since MN || BC, ∠MNC and ∠C are consecutive interior angles, so ∠MNC + ∠C = 180°.

(3y + 10) + (y + 10) = 180

4y + 20 = 180

4y = 160

y = 40

Step 3:

Since MN || BC, ∠AMN and ∠B are corresponding angles, so ∠AMN = ∠B.

2a + 15 = 3a – 15

a = 30

Step 4:

`MN = 1/2BC = 1/2(10) = 5  cm`

`MB = 1/2AB = 1/2(7) = 3.5  cm`

`NC = 1/2AC = 1/2(9.2) = 4.6  cm`

Perimeter of MNCB = MN + NC + CB + BM

Perimeter of MNCB = 5 + 4.6 + 10 + 3.5 = 23.1 cm

The values are x = 3.5, y = 40, a = 30 and the perimeter of MNCB is 23.1 cm.