SSC CGL Maths · Coordinate Geometry

Distance, Section Formula and Area

Work with points alone: how far apart two points are, the point halfway between them, the centroid of a triangle, and the area of a triangle from its three corners.

All Coordinate Geometry notes

Why this matters

Seven questions in the bank, five EASY and two MODERATE, all from the 2025 shifts. Four are about the midpoint or the centroid, often run backwards to find a missing corner. Each is one formula applied once, so the only real risk is a sign slip with a negative coordinate.

Concept 1 of 3: Distance between two points

Join the two points and drop a horizontal and a vertical line to make a right triangle. The across-gap and the up-gap are its two legs, and the distance is the hypotenuse, so it is Pythagoras on the coordinate grid.

Definition

The across-gap is $x_2 - x_1$ and the up-gap is $y_2 - y_1$ . Square each, add, take the square root.
The order of the points does not matter: squaring removes any minus sign.
From the origin to $(x, y)$ the distance is $\sqrt{x^2 + y^2}$ .
Fast route: if the two gaps are a known triple, read the answer. Gaps 3 and 4 give 5, 5 and 12 give 13, 8 and 15 give 17, 7 and 24 give 25, and any multiple of these works too (6 and 8 give 10). If the gaps match no triple, square and add.

Distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad d_{\text{origin}} = \sqrt{x^2 + y^2}

Worked example

Find the distance between

P(-2, 3)

and

Q(4, -5)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 1 · Q68Easy

Example 1 · Coordinate Geometry · Distance, Section Formula and Area

What is the length of the segment joining (1,2) and (4,6)?

Losing the minus sign when a coordinate is negative

The gap between

x = -2

and

x = 4

4 - (-2) = 6

, not 2. Write the subtraction with brackets whenever a coordinate is negative.

Adding the gaps instead of squaring them

Gaps of 3 and 4 give a distance of 5, not 7. The distance is the square root of the sum of the squares of the gaps, never the plain sum.

Concept 2 of 3: Midpoint, section formula and centroid

The midpoint is the average of the two ends, coordinate by coordinate. The centroid of a triangle is the average of its three corners. Both are averages, so a missing point comes back by multiplying out and subtracting.

Definition

Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ : average the x values and average the y values.
Section formula: the point dividing the segment in the ratio $m : n$ from the first point is a weighted average, with weight $m$ on the second point and $n$ on the first. The midpoint is the case $1 : 1$ .
Centroid of a triangle: the average of the three corners. It lies on each median and divides it $2 : 1$ from the corner, so with corner A and midpoint D of the opposite side, $G = \frac{A + 2D}{3}$ .
Missing corner, fast route: $x_3 = 3x_G - x_1 - x_2$ , and the same for $y$ . A missing end of a segment: $x_2 = 2x_M - x_1$ .
Right triangle: the point at the same distance from all three corners (the circumcentre) is the midpoint of the hypotenuse. This holds only when the triangle has a right angle.

Midpoint, section point and centroid

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \qquad P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right) \qquad G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)

m : nthe ratio in which P divides the segment, measured from the first point
Gthe centroid, where the three medians meet

Worked example

A triangle has corners

A(0, 3)

B(-3, 8)

and

C(9, -2)

. Find the midpoint M of BC and the centroid G of the triangle.

Practice this conceptself-check · 5 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 1 · Q73Easy

Example 2 · Coordinate Geometry · Distance, Section Formula and Area

In a triangle with vertices P(x, y), Q(5, -1), and R(2, 6), the Centroid is at G(4, 2). What are the coordinates of vertex P?

Dividing by 2 for a centroid

The centroid averages three corners, so the sum is divided by 3. Dividing by 2 gives a point that is not on any median.

Taking the centroid one third of the way from the corner

The centroid is two thirds of the way along a median from the corner, and one third of the way from the midpoint of the opposite side. The ratio from the corner is

2 : 1

Using the midpoint of the hypotenuse for any triangle

The point equidistant from three corners is the midpoint of the longest side only in a right triangle. Check first that two sides lie along perpendicular lines, such as the two axes.

Drill 4 more on midpoint, section formula and centroid

Concept 3 of 3: Area of a triangle from its corners

Three corners fix a triangle, so they fix its area. The formula is a criss-cross of the coordinates, and when the three points lie on one line the triangle is flat and the area comes out as zero.

Definition

Multiply each x by the difference of the other two y values, taken in turn, add the three products, take the absolute value and halve it.
Fast route: if one corner is the origin, the area is $\frac{1}{2}|x_1 y_2 - x_2 y_1|$ from the other two corners. If two sides lie along the axes, it is half the base times the height.
Collinear test: three points lie on one straight line exactly when this area is 0.

Area of a triangle from coordinates

\text{Area} = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|

Worked example

Find the area of the triangle with corners

(1, -2)

(5, 1)

and

(-1, 4)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 22 Sep 2025, Shift 3 · Q60Easy

Example 3 · Coordinate Geometry · Distance, Section Formula and Area

A triangle has vertices at

A(2,3)

B(7,1)

, and

C(4,6)

. What is its area?

Forgetting the half

The bracket gives twice the area. An option equal to the full bracket is usually printed to catch the student who stops before halving.

Reporting a negative area

Going round the corners in the other order flips the sign of the bracket. Area is never negative: take the absolute value before halving.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Distance between two points
Distance formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad d_{\text{origin}} = \sqrt{x^2 + y^2}$
Midpoint, section formula and centroid
Midpoint, section point and centroid
$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \qquad P = \left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right) \qquad G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$
Area of a triangle from its corners
Area of a triangle from coordinates
$\text{Area} = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$

Watch out for (7)

Losing the minus sign when a coordinate is negative
→ Distance between two points
Adding the gaps instead of squaring them
→ Distance between two points
Dividing by 2 for a centroid
→ Midpoint, section formula and centroid
Taking the centroid one third of the way from the corner
→ Midpoint, section formula and centroid
Using the midpoint of the hypotenuse for any triangle
→ Midpoint, section formula and centroid
Forgetting the half
→ Area of a triangle from its corners
Reporting a negative area
→ Area of a triangle from its corners

Test yourself on Coordinate Geometry

10 past SSC CGL questions from this chapter, timed at 6 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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Related notes

Triangles: centres of a triangle →