Question

M and N are mid-points of AB and AC.

1. Find x if ∠MNC = 3x – 10° and ∠C = x + 18°
2. Find y, if MN = (2y + 3) cm and BC = (3y + 8) cm

Answer 1

Given:

• M and N are mid-points of AB and AC respectively.
• ∠MNC = 3x – 10°
• ∠C = x + 18°
• MN = (2y + 3) cm
• BC = (3y + 8) cm

i. Find x

Step 1: Understanding the figure

Since M and N are midpoints of AB and AC, by the midpoint theorem, MN is parallel to BC and half of its length. Also, angles ∠MNC and ∠C are related.

Step 2: Using angle sum property

In triangle MNC,

• Angle at N is ∠MNC = 3x – 10°
• Angle at C is ∠C = x + 18°

Since M and N are mid-points, line MN is parallel to BC, so angle ∠MNC and ∠C are supplementary.

Thus,

∠MNC + ∠C = 180°
(3x – 10) + (x + 18) = 180
3x – 10 + x + 18 = 180
4x + 8 = 180
4x = 180 – 8
4x = 172
x = `172/4`
x = 43°

ii. Find y

From the Midpoint theorem,

MN = `1/2` × BC

Given:

MN = 2y + 3
BC = 3y + 8

So,
2y + 3 = `1/2` × (3y + 8)
Multiply both sides by 2:
2(2y + 3) = 3y + 8
4y + 6 = 3y + 8
4y – 3y = 8 – 6
y = 2