Coordinate Geometry Word Problems

Class 10Coordinate Geometry

Coordinate geometry word problems apply the distance formula, section formula, midpoint formula, and area formula to real-life and geometric situations. These problems are a key part of Class 10 NCERT Mathematics.

Students must translate a verbal description into coordinates, identify which formula to use, and compute the answer. Common scenarios include finding distances between landmarks, dividing paths in ratios, and computing areas of plots.

Mastering these problems requires fluency with all four core formulas and the ability to set up coordinate systems from word descriptions.

What is Coordinate Geometry Word Problems?

Core formulas used in coordinate geometry word problems:

Distance Formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Section Formula: The point dividing the join of (x₁, y₁) and (x₂, y₂) in ratio m:n is ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Area of Triangle: = (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Coordinate Geometry Word Problems Formula

Distance Formula:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Section Formula (Internal Division):

P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))

Area of Triangle:

A = (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Solved Examples

Example 1: Distance Between Two Locations

Problem: Two friends live at points A(2, 3) and B(6, 6) on a map where 1 unit = 1 km. Find the straight-line distance between their houses.

Solution:

Given: A(2, 3) and B(6, 6)

Using distance formula:

AB = √[(6−2)² + (6−3)²]
= √[16 + 9] = √25 = 5 km

Answer: The distance between their houses is 5 km.

Example 2: Finding a Point on a Road

Problem: A road runs from town A(1, 2) to town B(7, 8). A hospital is to be built at a point that divides the road in the ratio 2:1 from A. Find the location of the hospital.

Solution:

Given: A(1, 2), B(7, 8), ratio = 2:1

Using section formula:

x = (2×7 + 1×1)/(2+1) = (14+1)/3 = 15/3 = 5
y = (2×8 + 1×2)/(2+1) = (16+2)/3 = 18/3 = 6

Answer: The hospital should be at (5, 6).

Example 3: Finding the Centre of a Bridge

Problem: A bridge connects two points P(−3, 4) and Q(5, −2) across a river. Find the midpoint of the bridge.

Solution:

Using midpoint formula:

M = ((−3+5)/2, (4+(−2))/2)
= (2/2, 2/2) = (1, 1)

Answer: The centre of the bridge is at (1, 1).

Example 4: Area of a Triangular Plot

Problem: A farmer has a triangular plot with corners at A(2, 1), B(8, 3), and C(4, 7). Find the area of the plot if 1 unit = 10 m.

Solution:

Using the area formula:

Area = (1/2)|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
= (1/2)|2(3−7) + 8(7−1) + 4(1−3)|
= (1/2)|2(−4) + 8(6) + 4(−2)|
= (1/2)|−8 + 48 − 8|
= (1/2)|32| = 16 sq. units

Since 1 unit = 10 m, 1 sq. unit = 100 m².

Answer: Area = 16 × 100 = 1600 m².

Example 5: Equidistant Point on X-axis

Problem: Find the point on the x-axis that is equidistant from A(3, 4) and B(−1, 2).

Solution:

A point on the x-axis has coordinates (x, 0).

Setting distances equal:

PA = PB
√[(x−3)² + 16] = √[(x+1)² + 4]

Squaring both sides:

(x−3)² + 16 = (x+1)² + 4
x² − 6x + 9 + 16 = x² + 2x + 1 + 4
−6x + 25 = 2x + 5
20 = 8x
x = 5/2

Answer: The point is (5/2, 0) or (2.5, 0).

Example 6: Checking Collinearity of Three Locations

Problem: Three towns are located at A(1, 5), B(2, 3), and C(4, −1). Determine if they lie on a straight road.

Solution:

Three points are collinear if the area of the triangle they form is zero.

Area = (1/2)|1(3−(−1)) + 2(−1−5) + 4(5−3)|
= (1/2)|1(4) + 2(−6) + 4(2)|
= (1/2)|4 − 12 + 8|
= (1/2)|0| = 0

Answer: Since the area is 0, the three towns are collinear — they lie on a straight road.

Example 7: Ratio in Which a Point Divides a Line

Problem: The point C(4, 5) lies on the line segment joining A(2, 3) and B(8, 9). Find the ratio in which C divides AB.

Solution:

Let C divide AB in ratio k:1.

Using section formula (x-coordinate):

4 = (8k + 2)/(k + 1)
4(k + 1) = 8k + 2
4k + 4 = 8k + 2
2 = 4k → k = 1/2

Ratio = k:1 = 1/2 : 1 = 1:2

Verify with y-coordinate: (9×1/2 + 3)/(1/2 + 1) = (4.5 + 3)/1.5 = 7.5/1.5 = 5 ✓

Answer: C divides AB in the ratio 1:2.

Example 8: Trisection Points of a Line Segment

Problem: Find the trisection points of the line segment joining A(1, −2) and B(−3, 10).

Solution:

Trisection points divide the segment into three equal parts, at ratios 1:2 and 2:1.

Point P (ratio 1:2):

x = (1×(−3) + 2×1)/3 = (−3+2)/3 = −1/3
y = (1×10 + 2×(−2))/3 = (10−4)/3 = 2
P = (−1/3, 2)

Point Q (ratio 2:1):

x = (2×(−3) + 1×1)/3 = (−6+1)/3 = −5/3
y = (2×10 + 1×(−2))/3 = (20−2)/3 = 6
Q = (−5/3, 6)

Answer: The trisection points are (−1/3, 2) and (−5/3, 6).

Real-World Applications

Coordinate geometry word problems appear in:

Urban Planning: Finding optimal locations for hospitals, schools, and fire stations equidistant from multiple zones.
Navigation: Computing distances and midpoints between GPS coordinates.
Surveying: Calculating areas of land plots using coordinates of boundary markers.
Engineering: Placing structures at specific division points along a line.
Computer Graphics: Positioning objects on a 2D screen using coordinate systems.

Key Points to Remember

Always identify whether the problem requires distance, section, midpoint, or area formula.
A point on the x-axis has y = 0; a point on the y-axis has x = 0.
For equidistant problems, set the two distances equal and solve.
Three points are collinear if the area of the triangle = 0.
Trisection points divide a segment in ratios 1:2 and 2:1.
The section formula gives internal division; for external division, use (mx₂ − nx₁)/(m − n).
Midpoint is a special case of section formula with ratio 1:1.
Always verify your answer by substituting back into the original condition.
Convert units at the end if the problem uses a scale (e.g., 1 unit = 1 km).
For quadrilateral areas, split into two triangles and add the areas.

Practice Problems

Two schools are at A(−2, 3) and B(4, −1). A new bus stop is to be at the midpoint. Find its coordinates.
A park has corners at A(0, 0), B(6, 0), C(6, 4), and D(0, 4). Find the length of the diagonal path from A to C.
Find the point on the y-axis equidistant from A(5, 2) and B(−3, −2).
The coordinates of two opposite vertices of a rectangle are (2, 3) and (8, 7). Find the coordinates of the centre.
A triangular garden has vertices A(0, 0), B(10, 0), C(4, 6). Find its area in m² if 1 unit = 5 m.
In what ratio does the point (3, 4) divide the line joining (1, 2) and (7, 8)?

Frequently Asked Questions

Q1. How do you set up coordinates in a word problem when none are given?

Place one point at the origin (0, 0) and align one side along an axis. For example, if a field has a right angle at one corner, place that corner at the origin with sides along the x- and y-axes.

Q2. How do you find a point equidistant from two given points?

Let the unknown point be (x, y). Use the distance formula to set the two distances equal. If the point is on the x-axis, use (x, 0); if on the y-axis, use (0, y). Solve the resulting equation.

Q3. What is the difference between section formula and midpoint formula?

The midpoint formula is a special case of the section formula where the ratio is 1:1. Section formula works for any ratio m:n.

Q4. How do you check if three places are on the same straight road?

Use the area of triangle formula with the three points. If the area equals zero, the points are collinear (on the same straight line).

Q5. Can the section formula give a point outside the line segment?

The internal section formula gives a point between the two endpoints. For a point outside, use the external section formula: ((mx₂ − nx₁)/(m − n), (my₂ − ny₁)/(m − n)).

Q6. How do you find the area of a quadrilateral using coordinates?

Divide the quadrilateral into two triangles by drawing a diagonal. Compute the area of each triangle using the area formula and add them.

Coordinate Geometry Word Problems

What is Coordinate Geometry Word Problems?

Coordinate Geometry Word Problems Formula

Solved Examples

Example 1: Distance Between Two Locations

Example 2: Finding a Point on a Road

Example 3: Finding the Centre of a Bridge

Example 4: Area of a Triangular Plot

Example 5: Equidistant Point on X-axis

Example 6: Checking Collinearity of Three Locations

Example 7: Ratio in Which a Point Divides a Line

Example 8: Trisection Points of a Line Segment

Real-World Applications

Key Points to Remember

Practice Problems

Frequently Asked Questions

Q1. How do you set up coordinates in a word problem when none are given?

Q2. How do you find a point equidistant from two given points?

Q3. What is the difference between section formula and midpoint formula?

Q4. How do you check if three places are on the same straight road?

Q5. Can the section formula give a point outside the line segment?

Q6. How do you find the area of a quadrilateral using coordinates?

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