In a triangle $ABC,\text{ if }A=\left(3,2\right),B=\left(1,4\right)$  and the midpoint of $AC$  is $\left(2,5\right)$ , then midpoint of $BC$  is

In triangle ABC, the given points are:

• Point A = (3, 2)
• Point B = (1, 4)
• The midpoint of AC is D = (2, 5)

Let the coordinates of point C be C = (x, y).

Step 1: Using the midpoint formula for AC

The midpoint formula for a line segment connecting two points (x₁, y₁) and (x₂, y₂) is:

Midpoint = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

For triangle ABC, the midpoint of AC is given as D = (2, 5). Substituting the coordinates of A = (3, 2) and C = (x, y) into the formula:

(2, 5) = [(3 + x) / 2, (2 + y) / 2]

Step 2: Solving for x and y

Equating the respective coordinates:

• (3 + x) / 2 = 2 → 3 + x = 4 → x = 1
• (2 + y) / 2 = 5 → 2 + y = 10 → y = 8

Thus, the coordinates of point C in triangle ABC are C = (1, 8).

Step 3: Finding the midpoint of BC

Now, we calculate the midpoint of BC using the formula:

Midpoint = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

Substituting B = (1, 4) and C = (1, 8):

Midpoint of BC = [(1 + 1) / 2, (4 + 8) / 2]

Midpoint of BC = [2 / 2, 12 / 2] = (1, 6)

Final Answer

In triangle ABC, the midpoint of BC is (1, 6).