CBSE Class 10 Mathematics • Chapter 7
Coordinate Geometry Class 10 Worksheet with Solutions
Practise the distance formula, section formula and midpoint formula. Follow worked examples, solve a map-based case study and check your understanding with MCQs, extra questions and clear explanations.
This Coordinate Geometry Class 10 worksheet helps you connect algebra with geometry. Coordinates describe where a point lies, while formulas help calculate distances and locate points dividing a line segment. Careful substitution is essential, especially when coordinates contain negative numbers.
Use the worksheet available on this page alongside the original examples and revision material below. A worksheet on Coordinate Geometry Class 10 with answers is most useful when you attempt each question first, compare your method with the solution and then reattempt any incorrect calculation independently.
Coordinate Geometry Class 10 Introduction
Coordinate geometry represents points using ordered pairs (x, y) on the Cartesian plane. The x-coordinate gives the horizontal position, and the y-coordinate gives the vertical position. The axes intersect at the origin, (0, 0).
Basic Concepts to Remember
- A point on the x-axis has y-coordinate 0.
- A point on the y-axis has x-coordinate 0.
- The origin lies on both axes.
- The points (2, 5) and (5, 2) are different.
- Distance is non-negative, even when coordinates are negative.
| Quadrant | Signs of Coordinates | Example |
|---|---|---|
| I | (+, +) | (3, 4) |
| II | (−, +) | (−3, 4) |
| III | (−, −) | (−3, −4) |
| IV | (+, −) | (3, −4) |
Coordinate Geometry Class 10 Formula Sheet
Let A(x1, y1) and B(x2, y2) be two points. For internal division, let P divide AB in the ratio AP:PB = m:n, where m and n are positive.
| Formula | Expression | Use |
|---|---|---|
| Distance formula | AB = √[(x2 − x1)2 + (y2 − y1)2] | Find the length between two points. |
| Internal section formula | P = ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)) | Locate a point dividing a segment in a given ratio. |
| Midpoint formula | M = ((x1 + x2)/2, (y1 + y2)/2) | Divide a segment in the ratio 1:1. |
| Distance from the origin | OP = √(x2 + y2) | Find the distance of P(x, y) from (0, 0). |
Keep the Section Formula Ratio in the Correct Order
If AP:PB = m:n, multiply B’s coordinates by m and A’s coordinates by n. Write the endpoint labels and ratio before substituting. Reversing the ratio usually gives a different dividing point.
Coordinate Geometry Class 10 Questions with Solutions
Example 1: Distance Formula with Negative Coordinates
Question: Find the distance between A(−2, 3) and B(4, −5).
AB = √[(4 − (−2))2 + (−5 − 3)2]
AB = √[62 + (−8)2]
AB = √(36 + 64)
AB = √100
AB = 10 units
Example 2: Find a Midpoint
Question: Find the midpoint of the segment joining A(−4, 6) and B(8, −2).
M = ((−4 + 8)/2, (6 + (−2))/2)
M = (4/2, 4/2)
M = (2, 2)
The midpoint is obtained by averaging the x-coordinates and averaging the y-coordinates separately.
Example 3: Apply the Section Formula
Question: P divides the segment joining A(1, 2) and B(7, 8) internally in the ratio AP:PB = 1:2. Find P.
x = (1 × 7 + 2 × 1)/(1 + 2)
x = 9/3
x = 3
y = (1 × 8 + 2 × 2)/(1 + 2)
y = 12/3
y = 4
P = (3, 4). It is closer to A because AP is half of PB.
Example 4: Find a Point Equidistant from Two Points
Question: Find a point P on the x-axis that is equidistant from A(2, 3) and B(6, 5).
Since P lies on the x-axis, write P = (x, 0). Equal distances imply equal squared distances.
(x − 2)2 + 32 =
(x − 6)2 + 52
x2 − 4x + 13 =
x2 − 12x + 61
8x = 48
x = 6
P = (6, 0).
PA = √(16 + 9) = 5 units.
PB = √(0 + 25) = 5 units.
Coordinate Geometry Class 10 Case Study Questions
Case Study: A School Campus Map
On a coordinate map, the school gate is A(2, 2) and the library is B(10, 8). One coordinate unit represents 10 metres. A drinking-water station P lies on the straight segment AB such that AP:PB = 1:3.
1. Find the straight-line distance from the gate to the library.
AB = √[(10 − 2)2 + (8 − 2)2]
AB = √(64 + 36)
AB = 10 coordinate units
Actual distance = 100 metres
2. Find the midpoint of AB.
M = ((2 + 10)/2, (2 + 8)/2)
M = (6, 5)
3. Find the drinking-water station’s coordinates.
x = (1 × 10 + 3 × 2)/4
x = 4
y = (1 × 8 + 3 × 2)/4
y = 3.5
P = (4, 3.5)
4. How far is P from the gate?
AP is one-fourth of AB.
AP = 25 metres
Additional Coordinate Geometry Formulas
Some textbooks and older exercise sets also include area and centroid questions. Use the following results when they are part of your assigned course.
Area of a Triangle
For A(x1, y1), B(x2, y2) and C(x3, y3):
Area = ½ × absolute value of [x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)]
The absolute value makes the area non-negative. Three distinct points with area zero are collinear.
Centroid of a Triangle
G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
The centroid is the intersection of the three medians.
Coordinate Geometry Class 10 MCQ with Answers
1. What is the distance between (0, 0) and (3, 4)?
A. 3 B. 4 C. 5 D. 7
Answer: C. 5 units. √(32 + 42) = √25 = 5.
2. What is the midpoint of (−2, 4) and (6, 8)?
A. (2, 6)
B. (4, 12)
C. (−4, −2)
D. (6, 2)
Answer: A. (2, 6). Average the corresponding coordinates.
3. Which coordinate must be zero for a point on the y-axis?
A. x-coordinate B. y-coordinate C. Both always
Answer: A. Every point on the y-axis has the form (0, y).
4. In what ratio does a midpoint divide a segment?
A. 1:2 B. 2:1 C. 1:1 D. 1:3
Answer: C. 1:1. The two parts have equal lengths.
Coordinate Geometry Class 10 Extra Questions
- Find the distance between (−1, 2) and (5, 10).
- Find the midpoint of (3, −5) and (−7, 9).
- Find P dividing A(2, 1) and B(8, 7) internally in the ratio AP:PB = 2:1.
- One endpoint of a segment is (1, 3) and its midpoint is (4, 5). Find the other endpoint.
- Determine whether A(0, 0), B(3, 4) and C(6, 8) are collinear using their distances.
Check the Practice Answers
- 10 units.
- (−2, 2).
- P = (6, 5).
- (7, 7).
- AB = 5, BC = 5 and AC = 10. Since AB + BC = AC, the points are collinear, with B between A and C.
Using This Worksheet with NCERT Exercises and Board Questions
For Coordinate Geometry Class 10 Exercise 7.1, practise distance formula questions and geometric conclusions based on lengths. For Exercise 7.2, focus on internal division, midpoint questions and finding unknown ratios. Follow the exercise content in your own NCERT edition.
Use these examples alongside Coordinate Geometry Class 10 NCERT solutions 7.1 and 7.2, NCERT Exemplar problems and verified board papers. When selecting a Coordinate Geometry Class 10 previous year questions PDF, check the board and year. The examples on this page are original practice questions.
A Coordinate Geometry Class 10 test paper PDF with solutions can help you practise under timed conditions. After checking your answers, identify whether an error came from choosing the formula, reversing a ratio or handling a negative coordinate. Reattempt the question after revising that specific skill.
Common Mistakes to Avoid
- Writing x and y in the wrong order.
- Forgetting brackets when subtracting negative coordinates.
- Leaving out the square root in the distance formula.
- Reversing AP:PB in the section formula.
- Using the midpoint formula for a ratio other than 1:1.
- Confusing coordinate units with the actual map scale.
FAQs on Coordinate Geometry Class 10 Worksheet
What is coordinate geometry?
Coordinate geometry studies geometric figures using coordinates and algebra. For example, the distance formula calculates the length between two points without measuring a drawing.
How is the distance formula derived?
The horizontal and vertical differences between two points form the perpendicular sides of a right triangle. Applying Pythagoras’ theorem gives the squared distance as (x2 − x1)2 + (y2 − y1)2. Taking the square root gives the distance.
When should I use the section formula instead of the midpoint formula?
Use the section formula for a specified internal ratio m:n. Use the midpoint formula when the ratio is 1:1. The midpoint formula is a special case of the section formula.
When do I use k:1 in section formula questions?
Use k:1 when an internal division ratio is unknown. Substitute the dividing point’s coordinates into the section formula and solve for k. Use 1:1 only when the point is known to be the midpoint.
How do I find the ratio in which the x-axis divides a segment?
Set the dividing point’s y-coordinate equal to zero. For example, with A(2, −3), B(8, 6) and AP:PB = m:n, (6m − 3n)/(m + n) = 0 gives m:n = 1:2. The point of division is (4, 0).
Can a distance be negative?
No. Distance is non-negative. The coordinate differences may be negative, but they are squared before taking the non-negative square root. Distance is zero only when the two points coincide.
How can I prove that a triangle is right-angled using coordinates?
Calculate the three squared side lengths. If the largest squared length equals the sum of the other two, the triangle is right-angled by the converse of Pythagoras’ theorem. Check that the three vertices are distinct.
How can I prove that three points are collinear?
One method is to show that the sum of two distances equals the third, placing one point between the other two. Where included in your course, the area formula also gives a test: three distinct points are collinear when their triangle’s area is zero.
What are some coordinate geometry in real life examples?
Examples include locating places on a grid map, positioning objects in a design and calculating straight-line distances on a campus plan. Use the given scale to convert coordinate units into actual distances.
Which Coordinate Geometry Class 10 important questions should I practise?
Practise distances, midpoints, internal division, unknown endpoints, equidistant points, collinearity and identifying shapes from side lengths. Include case-based questions that require interpreting coordinates and units.
Are Maharashtra Board Practice Set 5.1 and CBSE Exercise 7.1 the same?
No. Their textbook questions and numbering differ. For SSC Practice Sets 5.1, 5.2, 5.3 and Problem Set 5, use the Maharashtra Board Coordinate Geometry Solutions . For CBSE, follow Chapter 7 in your NCERT textbook.
How should I use a Coordinate Geometry Class 10 worksheet PDF?
Attempt the questions first, then check the solutions. Keep the formula, substitution and calculation on separate lines. Reattempt incorrect questions without looking at the answer.
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