Find Triangle Vertices from Mid-points of Sides | Class 10

Quick answer: If D, E and F are the mid-points of BC, CA and AB of triangle ABC, then each vertex is the sum of the two adjacent mid-points minus the opposite one: $A = E + F - D$, $B = F + D - E$, $C = D + E - F$ (apply to x and y separately). For mid-points D(3, 4), E(4, 6) and F(5, 7), the vertices are A(6, 9), B(4, 5) and C(2, 3).

The setup

Let the vertices be $A(x_1, y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$, and let:

• D be the mid-point of BC,
• E be the mid-point of CA,
• F be the mid-point of AB.

         A
        / \
       F---E
      / \ / \
     B---D---C

Method 1: using the mid-point formula

Step 1: Write three equations for the x-coordinates.

$x_2 + x_3 = 2x_D \qquad \text{(1)}$

$x_3 + x_1 = 2x_E \qquad \text{(2)}$

$x_1 + x_2 = 2x_F \qquad \text{(3)}$

Step 2: Add all three.

$2(x_1 + x_2 + x_3) = 2(x_D + x_E + x_F)$

$x_1 + x_2 + x_3 = x_D + x_E + x_F$

Step 3: Subtract each equation from this sum.

$x_1 = x_E + x_F - x_D$

$x_2 = x_F + x_D - x_E$

$x_3 = x_D + x_E - x_F$

The same results hold for the y-coordinates.

Rule: For each vertex, add the two mid-points next to it and subtract the mid-point opposite it.

Method 2: parallelogram check

By the mid-point theorem (Class 9), FE ∥ BC and FD ∥ AC, so AFDE is a parallelogram. The diagonals of a parallelogram bisect each other, so the mid-point of AD equals the mid-point of EF:

$\frac{A + D}{2} = \frac{E + F}{2} \quad \Rightarrow \quad A = E + F - D$

This confirms the rule from Method 1.

Solved example

The mid-points of the sides BC, CA and AB of a triangle are D(3, 4), E(4, 6) and F(5, 7). Find the vertices.

Step 1: Vertex A (add E and F, subtract D):

$A = (4 + 5 - 3,\ 6 + 7 - 4) = (6, 9)$

Step 2: Vertex B (add F and D, subtract E):

$B = (5 + 3 - 4,\ 7 + 4 - 6) = (4, 5)$

Step 3: Vertex C (add D and E, subtract F):

$C = (3 + 4 - 5,\ 4 + 6 - 7) = (2, 3)$

Step 4: Check.

Answer: A(6, 9), B(4, 5), C(2, 3).

Summary table

Exam tips

• Write the three mid-point equations first; they earn method marks.
• Be clear about which mid-point lies on which side before you start.
• Always check by finding the mid-points of your answer.
• The triangle and its mid-point triangle have the same sum of coordinates, which is a quick check.

Common mistakes

• Subtracting the wrong mid-point. Subtract the one opposite the vertex.
• Mixing up which side each mid-point belongs to.
• Forgetting to do the y-coordinates separately.
• Halving instead of doubling when rearranging the mid-point formula.

Frequently Asked Questions

How do you find the vertices of a triangle from the mid-points of its sides?

Add the coordinates of the two mid-points next to a vertex and subtract the mid-point opposite it. Repeat for each vertex and for both x and y.

What are the vertices if the mid-points are D(3, 4), E(4, 6) and F(5, 7)?

The vertices are A(6, 9), B(4, 5) and C(2, 3), where D, E and F are the mid-points of BC, CA and AB.

Why does the rule A = E + F − D work?

AFDE is a parallelogram, and its diagonals AD and EF bisect each other, so A + D = E + F.

What is the mid-point formula?

The mid-point of (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

How can I check my answer?

Find the mid-points of the sides of your triangle. They should match the given mid-points.