1
(−4, 0) lies on —
💡 Hint:
The second number is 0.
Explanation:
y = 0, so the x-axis.
Class 10 · Maths · Chapter 7 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Coordinate Geometry
Coordinate Geometry
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- 1 Locating a point on the plane
- 2 The distance formula
- 3 Collinear points and the type of triangle
- 4 A point the same distance away
- 5 The internal section formula
- 6 Midpoint, trisection and diagonals
- Chapter winner — every lesson at mastery ★
🎬 Shorts
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📄 PDF download — notes + answer key
Locating a point on the plane
तल पर बिंदु का पता · NCERT 7.1 · abscissa x · ordinate y
Two numbers for one pointNotes
A place on the plane needs two perpendicular axes. The distance from the y-axis is the abscissa (x). The distance from the x-axis is the ordinate (y).
Every point on the x-axis is (x, 0). Every point on the y-axis is (0, y). The origin is (0, 0). Two numbers without order do not make one point: (2, 5) and (5, 2) are different places.
Draw the axes on paper and join the pointsActivity
Section 7.1 asks you to build a picture. Draw perpendicular axes on graph paper. Join the given points in order. See which points sit on the x-axis. The second number there is 0. On the y-axis the first number is 0.
Worked exampleExample
Question: Which axis holds A(4, 0), B(0, 7) and C(−3, 5)?
Formula: If y = 0, the point is on the x-axis. If x = 0, it is on the y-axis. If neither is zero, it is on no axis.
Substitution: For A, y = 0. For B, x = 0. For C, x = −3 and y = 5.
Answer: A is on the x-axis. B is on the y-axis. C is on neither axis.
10-second revision
- In (x, y), x is the abscissa and y the ordinate
- On the x-axis the ordinate is 0
- (2, 5) and (5, 2) are not one point
Board tip · BSEBBoard tip
In a BSEB answer write both words, abscissa and ordinate. Only x and y is thin.
Board tip · CBSEBoard tip
On CBSE name the axis only when one coordinate really is zero.
Check your understandingall correct = mastery ★
2
(3, 5) and (5, 3) are the same point.
Explanation:
False — changing the order changes the place.
3
The abscissa of a point on the y-axis is ______.
💡 Hint:
The first number.
Explanation:
0. The form is (0, y).
4
The ordinate is measured —
💡 Hint:
y is up or down from the x-axis.
Explanation:
The ordinate is the distance from the x-axis. The abscissa is the distance from the y-axis.
5
Which axis holds the origin and the point (0, −6)? Give the reason in one line.
Model answer (board format)
Both have abscissa 0, so both lie on the y-axis. The origin is also the crossing of the two axes.
Check your answer:
The distance formula
दूरी सूत्र · NCERT 7.2 · Exercise 7.1 · questions 1 and 2
Distance from PythagorasNotes
Between P(x1, y1) and Q(x2, y2) the horizontal step is x2 − x1 and the vertical step is y2 − y1. PQ = √[(x2 − x1)² + (y2 − y1)²]
Because of the squares, the distance is the same if (x1, y1) is written first or (x2, y2). The negative square root is not a length. The first questions of exercise 7.1 ask for the distance of a pair. Keep the unit that the axes use.
The two differences first, then the squares, then the rootActivity
Take (1, 2) and (4, 6). The differences are 4 − 1 = 3 and 6 − 2 = 4. The squares are 9 and 16. The sum is 25. The positive root is 5. If both points lie on one horizontal line, the vertical difference is 0 and the distance is only |x2 − x1|.
Worked exampleExample
Question: Find the distance between (1, 2) and (4, 6).
Formula: PQ = √[(x2 − x1)² + (y2 − y1)²].
Substitution: √[(4 − 1)² + (6 − 2)²] = √[9 + 16] = √25.
Answer: 5 units. Check: a 3-4-5 right triangle.
10-second revision
- Distance = √[(Δx)² + (Δy)²]
- Only the positive root is a length
- On one horizontal line the distance is |Δx|
Board tip · BSEBBoard tip
In a BSEB answer write the differences, the sum of squares and the root on three lines.
Board tip · CBSEBoard tip
On CBSE write √25 as 5, and do not call −5 a distance.
Check your understandingall correct = mastery ★
1
The distance between (1, 2) and (4, 6) is —
💡 Hint:
3² + 4² = 25.
Explanation:
√25 = 5 units.
2
The distance from (4, 6) to (1, 2) differs from the reversed pair.
Explanation:
False — the square removes the sign of the difference.
3
The distance between (0, 0) and (6, 8) is ______ units.
💡 Hint:
36 + 64 = 100.
Explanation:
√100 = 10 units.
4
Find the distance between (2, 5) and (2, −2).
Model answer (board format)
Δx = 0 and Δy = −2 − 5 = −7. The distance is √[0 + 49] = 7 units. On one vertical line this is |Δy|.
Check your answer:
Collinear points and the type of triangle
संरेख बिंदु और त्रिभुज का प्रकार · Exercise 7.1 · questions 3 to 6
Three distances decide itNotes
Find the three distances of three points. If the two smaller distances add up to the largest, the points are collinear. If the sum is larger, they form a triangle.
Two equal sides mean isosceles. All three equal means equilateral. If the square of the longest side equals the sum of the squares of the other two, it is right-angled. Questions 3 to 6 of exercise 7.1 are these checks. A square needs four equal sides and two equal diagonals.
Keep the largest distance on one sideActivity
The distances of (0, 0), (1, 1) and (2, 2) are √2, √2 and 2√2. √2 + √2 = 2√2, so they are collinear. For (0, 0), (4, 0) and (2, 3) the sides are 4, √13 and √13. √13 + √13 > 4, so it is a triangle, and two sides are equal, so it is isosceles.
Worked exampleExample
Question: What type of triangle do (0, 0), (4, 0) and (2, 3) form?
Formula: Distance √[(Δx)² + (Δy)²], then compare the sides.
Substitution: The base is √[(4 − 0)² + 0] = 4. The other two are √[(2 − 0)² + (3 − 0)²] = √13 and √[(2 − 4)² + (3 − 0)²] = √13.
Answer: Two sides are √13 units, so the triangle is isosceles. It is not equilateral, because the base is 4.
10-second revision
- Collinear: smaller + smaller = largest
- Two equal sides mean isosceles
- For a square check both sides and diagonals
Board tip · BSEBBoard tip
In a BSEB answer write all three distances as numbers, then the line that adds them.
Board tip · CBSEBoard tip
On CBSE do not call a triangle isosceles because the roots look close. The square roots must be exactly equal.
Check your understandingall correct = mastery ★
1
(0, 0), (1, 1), (2, 2) are —
💡 Hint:
√2 + √2 = 2√2.
Explanation:
The distances add up to the largest, so they are collinear.
2
(0, 0), (4, 0) and (2, 3) form an equilateral triangle.
Explanation:
False — two sides are √13 and the base is 4. It is isosceles.
3
The distance between (0, 0) and (4, 0) is ______ units.
💡 Hint:
Only the x difference.
Explanation:
4 units.
4
This is not enough before you call it a square —
💡 Hint:
A rhombus also has equal sides.
Explanation:
The diagonals must be equal along with the sides.
5
(1, 1), (1, 3), (3, 3), (3, 1) are the vertices of a square.
Explanation:
True — every side is 2 units and both diagonals are 2√2 units.
6
Are (0, 0), (3, 0) and (0, 4) a right triangle? Show the distances.
Model answer (board format)
The sides are 3, 4 and 5 units. 3² + 4² = 9 + 16 = 25 = 5². The angle opposite the longest side is a right angle.
Check your answer:
A point the same distance away
समदूरस्थ बिंदु · Exercise 7.1 · questions 7 to 10
Set the two distances equalNotes
A point on the x-axis is (x, 0). If it is equidistant from (x1, y1) and (x2, y2), then (x − x1)² + (0 − y1)² = (x − x2)² + (0 − y2)².
If y is unknown and the distance is given, then (Δx)² + (Δy)² = (distance)². A square equation can have two roots. Write both, unless the question asks for only one. The last questions of exercise 7.1 are this.
Write the point on the axis, then build the equationActivity
The point on the x-axis equidistant from (2, −5) and (−2, 9) is (x, 0). (x − 2)² + 25 = (x + 2)² + 81. Open the squares. x² cancels. −4x + 29 = 4x + 85. x = −7. The point is (−7, 0). The distance on both sides is √106 units.
Worked exampleExample
Question: The distance from P(2, −3) to Q(10, y) is 10 units. Find y.
Formula: (x2 − x1)² + (y2 − y1)² = (distance)².
Substitution: (10 − 2)² + (y + 3)² = 10². 64 + (y + 3)² = 100. (y + 3)² = 36.
Answer: y + 3 = 6 or y + 3 = −6. So y = 3 or y = −9. Both distances are 10 units.
10-second revision
- On the x-axis write the point (x, 0)
- Equidistant means the squares of both distances are equal
- Keep both ± roots when both give the distance
Board tip · BSEBBoard tip
In a BSEB answer show the line after x² cancels. Do not jump to x = −7.
Board tip · CBSEBoard tip
On CBSE one root is incomplete when (y + 3)² = 36.
Check your understandingall correct = mastery ★
1
The point on the x-axis equidistant from (2, −5) and (−2, 9) is —
💡 Hint:
(x − 2)² + 25 = (x + 2)² + 81.
Explanation:
x = −7. The point is (−7, 0).
2
There is only one value of y for which Q(10, y) is 10 units from P(2, −3).
Explanation:
False — both y = 3 and y = −9 work.
3
The distance of (3, 4) from the origin is ______ units.
💡 Hint:
√(9 + 16).
Explanation:
5 units.
4
The distance of (5, 12) from the origin is —
💡 Hint:
25 + 144 = 169.
Explanation:
√169 = 13 units.
5
Find the point on the x-axis equidistant from (0, 4) and (6, −2).
Model answer (board format)
The point is (x, 0). (x − 0)² + (0 − 4)² = (x − 6)² + (0 + 2)². x² + 16 = x² − 12x + 36 + 4. 12x = 24. x = 2. The point is (2, 0). Both distances are √20 units.
Check your answer:
The internal section formula
आंतरिक विभाजन सूत्र · NCERT 7.3 · Exercise 7.2 · questions 1, 4, 5
m1 goes with the second pointNotes
P divides A(x1, y1) and B(x2, y2) internally in m1 : m2, with AP : PB = m1 : m2. x = (m1 x2 + m2 x1) / (m1 + m2), y = (m1 y2 + m2 y1) / (m1 + m2)
m1 goes with the numbers of B. Reversing the ratio moves the point. This book does not teach external division. The opening questions of exercise 7.2 are internal ratios. A point that cuts the x-axis has ordinate 0.
Tie the parts of the ratio to the lettersActivity
A(2, 3), B(8, 15), ratio 1 : 2. m1 = 1, with B. m2 = 2, with A. x = (1×8 + 2×2) / 3 = 4. y = (1×15 + 2×3) / 3 = 7. The point is (4, 7). Check: AP = 2√5 and PB = 4√5, ratio 1 : 2.
Worked exampleExample
Question: In what ratio does the x-axis cut the segment joining A(1, −2) and B(4, 4)? Find the point too.
Formula: y = (m1 y2 + m2 y1) / (m1 + m2) = 0.
Substitution: (m1×4 + m2×(−2)) / (m1 + m2) = 0. 4 m1 = 2 m2. m1 : m2 = 1 : 2. x = (1×4 + 2×1) / 3 = 2.
Answer: The ratio is 1 : 2. The point is (2, 0). The ordinate is 0, so it lies on the x-axis.
10-second revision
- AP : PB = m1 : m2
- m1 with B and m2 with A
- External division is not in this chapter
Board tip · BSEBBoard tip
In a BSEB answer write both the word “internal” and the line AP : PB.
Board tip · CBSEBoard tip
On CBSE, swapping m1 and m2 is a common miss. Do not read the ratio 2 : 1 as 1 : 2.
Check your understandingall correct = mastery ★
1
The point dividing A(2, 3) and B(8, 15) in 1 : 2 is —
💡 Hint:
m1 = 1, with B.
Explanation:
x = 12/3 = 4, y = 21/3 = 7.
2
The section formula of this chapter is also for an external point.
Explanation:
False — the book gives internal division only.
3
The segment A(1, −2) to B(4, 4) meets the x-axis in the ratio ______.
💡 Hint:
4 m1 = 2 m2.
Explanation:
1 : 2. The point is (2, 0).
4
In what ratio does (3, 4) divide A(1, 2) and B(7, 8)?
Model answer (board format)
Let AP : PB = k : 1. (7k + 1) / (k + 1) = 3. 7k + 1 = 3k + 3. 4k = 2. k = 1/2. The ratio is 1 : 2. Check in y: (1×8 + 2×2) / 3 = 4.
Check your answer:
Midpoint, trisection and diagonals
मध्यबिंदु, त्रिभाजन और विकर्ण · Exercise 7.2 · question 2 and questions 6 to 10
The division 1 : 1 is the midpointNotes
The midpoint of P(x1, y1) and Q(x2, y2) is ((x1 + x2)/2, (y1 + y2)/2).
The trisection points come from the ratios 1 : 2 and 2 : 1. Four equal parts use 1 : 3, 1 : 1 and 3 : 1. In a parallelogram the diagonals share one midpoint. The centre of the diameter AB is that midpoint. The rhombus question in exercise 7.2 asks for the lengths of the diagonals. Area = (1/2) × diagonal₁ × diagonal₂.
Find the midpoint on each diagonal separatelyActivity
If the vertices in order are (1, 2), (4, y), (x, 6), (3, 5), the diagonals run from (1, 2) to (x, 6) and from (4, y) to (3, 5). The midpoints are ((1 + x)/2, 4) and (7/2, (y + 5)/2). Setting them equal gives x = 6 and y = 3. If a diameter gives the centre, the same addition runs backwards.
| Ask | Ratio | Example |
|---|---|---|
| Midpoint | 1 : 1 | Of (0, 0) and (8, 0): (4, 0) |
| Trisection | 1 : 2 and 2 : 1 | (3, 0) and (6, 0) |
| Four parts | 1 : 3, 1 : 1, 3 : 1 | (2, 0), (4, 0), (6, 0) |
Worked exampleExample
Question: The centre of a circle is (2, −3) and one end of a diameter is B(1, 4). Find the other end A.
Formula: Centre = ((x + 1)/2, (y + 4)/2).
Substitution: (x + 1)/2 = 2, so x + 1 = 4. (y + 4)/2 = −3, so y + 4 = −6.
Answer: A(3, −10). Check: the midpoint ((3 + 1)/2, (−10 + 4)/2) = (2, −3).
10-second revision
- The midpoint is the average of the coordinates
- Trisection uses 1 : 2 and 2 : 1
- A rhombus area is half the product of the diagonals
Board tip · BSEBBoard tip
In a BSEB diameter question write the midpoint check on the last line.
Board tip · CBSEBoard tip
On CBSE write both trisection points. Only the midpoint is incomplete.
Check your understandingall correct = mastery ★
1
The midpoint of (2, −4) and (8, 6) is —
💡 Hint:
Take the averages.
Explanation:
((2+8)/2, (−4+6)/2) = (5, 1).
2
The trisection points of (0, 0) and (9, 0) are (3, 0) and (6, 0).
Explanation:
True — the ratios 1 : 2 and 2 : 1.
3
If the centre is (2, −3) and B is (1, 4), the other end of the diameter is ______.
💡 Hint:
The midpoint run backwards.
Explanation:
A(3, −10).
4
In the parallelogram (1, 2), (4, y), (x, 6), (3, 5), x and y are —
💡 Hint:
One midpoint of the diagonals.
Explanation:
x = 6 and y = 3.
5
The area of a rhombus is half the product of its diagonals.
Explanation:
True — the hint in exercise 7.2 says this.
6
Find the area of the rhombus with vertices (2, 0), (0, 2), (−2, 0), (0, −2).
Model answer (board format)
The diagonal from (2, 0) to (−2, 0) is 4 units, and from (0, 2) to (0, −2) is 4 units. Area = (1/2) × 4 × 4 = 8 square units.
Check your answer:
❓ Full question bank — with answers and explanations — 65 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
A point on the y-axis has the form —
💡 Hint:
The abscissa is zero.
Explanation:
(0, y).
2
The distance between (1, 2) and (4, 6) is —
💡 Hint:
9 + 16.
Explanation:
5 units.
3
(0, 0), (1, 1), (2, 2) —
💡 Hint:
√2 + √2 = 2√2.
Explanation:
They are collinear.
4
The distance of (3, 4) from the origin is —
💡 Hint:
√(9+16).
Explanation:
5 units.
5
The x-axis point equidistant from (2, −5) and (−2, 9) is —
💡 Hint:
Set the squares equal.
Explanation:
(−7, 0).
6
For A(2, 3), B(8, 15), ratio 1 : 2, the point is —
💡 Hint:
m1 with the second point.
Explanation:
(4, 7).
7
The midpoint of (2, −4) and (8, 6) is —
💡 Hint:
Averages.
Explanation:
(5, 1).
8
If the centre is (2, −3) and one end is (1, 4), the other end is —
💡 Hint:
Midpoint backwards.
Explanation:
(3, −10).
9
(0, 0), (4, 0) and (2, 3) form —
💡 Hint:
Two sides are √13.
Explanation:
An isosceles triangle.
10
If Q(10, y) is 10 units from P(2, −3), y is —
💡 Hint:
(y+3)² = 36.
Explanation:
y = 3 or y = −9.
11
The x-axis cuts A(1, −2) and B(4, 4) in —
💡 Hint:
Ordinate 0.
Explanation:
Ratio 1 : 2 and the point (2, 0).
12
The area of the rhombus (2, 0), (0, 2), (−2, 0), (0, −2) is —
💡 Hint:
(1/2)×4×4.
Explanation:
8 square units.
True or false
0/81
(4, 0) lies on the x-axis.
Explanation:
True — the ordinate is 0.
2
The negative square root is also a valid length.
Explanation:
False — a distance is positive.
3
(1, 1), (1, 3), (3, 3), (3, 1) is a square.
Explanation:
True — side 2, diagonal 2√2.
4
This chapter has the formula for external division.
Explanation:
False — only internal division.
5
The midpoint is the ratio 1 : 1.
Explanation:
True.
6
(0, 0), (3, 0), (0, 4) are collinear.
Explanation:
False — the sides are 3, 4, 5, a right triangle.
7
In the section formula m1 is multiplied with the first point A.
Explanation:
False — m1 goes with B.
8
The distance of (5, 12) from the origin is 13 units.
Explanation:
True — 25 + 144 = 169.
Fill in the blanks
0/81
The abscissa of (0, 5) is ______.
💡 Hint:
0.
Explanation:
The y-axis.
2
The distance between (0, 0) and (6, 8) is ______ units.
💡 Hint:
10 units.
Explanation:
√100.
3
The midpoint y-coordinate of (2, −4) and (8, 6) is ______.
💡 Hint:
1.
Explanation:
(−4+6)/2.
4
Dividing A(2, 3) and B(8, 15) in 1 : 2 gives x = ______.
💡 Hint:
4.
Explanation:
(8+4)/3.
5
The distance of (3, 4) from the origin is ______ units.
💡 Hint:
5 units.
Explanation:
3-4-5.
6
If the centre is the midpoint, one end is (1, 4) and the centre is (2, −3), the other end has x = ______.
💡 Hint:
3.
Explanation:
(x+1)/2 = 2.
7
One trisection point of (0, 0) and (9, 0) is (3, ______).
💡 Hint:
0.
Explanation:
Both y values are zero.
8
A rhombus with diagonals 4 and 4 has area ______ square units.
💡 Hint:
8.
Explanation:
Half the product.
Match
0/21
Match the name with the formula.
Column B: A. ((x1+x2)/2, (y1+y2)/2) · B. √[(x2−x1)²+(y2−y1)²] · C. √(x²+y²) · D. (m1x2+m2x1)/(m1+m2)
1. Distance
2. Distance from origin
3. Section
4. Midpoint
Explanation:
Distance uses both differences, origin distance is √(x²+y²), section is a weighted average, and the midpoint is the plain average.
2
Match the part of the exercise with its job.
Column B: A. The centre is the midpoint · B. Distance of two points · C. Smaller + smaller = largest · D. m1 with B
1. Opening questions of 7.1
2. Collinear check in 7.1
3. Internal ratio in 7.2
4. Diameter in 7.2
Explanation:
7.1 is distance and collinear points. 7.2 is section and the midpoint.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): The distance between (1, 2) and (4, 6) is 5 units.
Reason (R): The distance is √[(Δx)² + (Δy)²] and 3² + 4² = 25.
Explanation:
Both are true and R is the basis of the calculation.
2
Assertion (A): (4, 0) lies on the x-axis.
Reason (R): The midpoint is the average of the coordinates.
Explanation:
Both are true, but R does not explain why the point is on the axis.
3
Assertion (A): (0, 0), (4, 0), (2, 3) is an isosceles triangle.
Reason (R): All three sides are 4 units.
Explanation:
A is true. R is false, only two sides are √13.
4
Assertion (A): This chapter teaches the external section formula.
Reason (R): In the internal formula m1 goes with the coordinates of the second point.
Explanation:
A is false. R is true.
5
Assertion (A): If the centre is (2, −3) and one end is (1, 4), the other end is (3, −10).
Reason (R): The centre is the midpoint of the diameter.
Explanation:
Both are true and R is the reason.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each set of points.
(4, 0)
(0, −6)
(−3, 5)
(0, 0)
Explanation:
Ordinate 0 means the x-axis. Abscissa 0 means the y-axis. The origin is the crossing of both.
2
Place each job with its tool.
How far two points are
Cutting a segment in 1 : 1
Whether three points lie on one line
The other end of a diameter
Explanation:
Distance measures lengths, the midpoint halves, collinear points show up in the sum, and a diameter runs the midpoint backwards.
Very short answer
0/51
Write the distance formula.
Model answer (board format)
The distance between P(x1, y1) and Q(x2, y2) is √[(x2−x1)² + (y2−y1)²].
Check your answer:
2
Write the distance of (x, y) from the origin.
Model answer (board format)
√(x² + y²).
Check your answer:
3
Write x for an internal division.
Model answer (board format)
x = (m1 x2 + m2 x1) / (m1 + m2), where AP : PB = m1 : m2.
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4
Write the midpoint formula.
Model answer (board format)
((x1 + x2)/2, (y1 + y2)/2).
Check your answer:
5
Separate abscissa and ordinate in one line.
Model answer (board format)
The abscissa is x, the distance from the y-axis. The ordinate is y, the distance from the x-axis.
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Short answer
0/41
Find the distance between (−1, 4) and (2, 8).
Model answer (board format)
Δx = 3, Δy = 4. Distance √[9 + 16] = √25 = 5 units.
Check your answer:
2
Find the point dividing A(0, 0) and B(6, 3) in 1 : 2.
Model answer (board format)
x = (1×6 + 2×0)/3 = 2. y = (1×3 + 2×0)/3 = 1. The point is (2, 1).
Check your answer:
3
Find the midpoint of (−2, 5) and (4, −1).
Model answer (board format)
((−2+4)/2, (5+(−1))/2) = (1, 2).
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4
Are (0, 0), (6, 0), (3, 3) collinear or a triangle? Write the sides.
Model answer (board format)
The sides are 6, √(9+9)=3√2 and 3√2 units. 3√2 + 3√2 = 6√2 > 6, so it is a triangle, and isosceles.
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Long answer
0/31
Find the point on the x-axis equidistant from (3, 4) and (3, −2). If none exists, say why. Then find the point for the pair (0, 4) and (6, −2).
Model answer (board format)
The first pair lies on the vertical line x = 3. (x−3)² + 16 = (x−3)² + 4 is impossible, so no such point. Second pair: (x−0)² + 16 = (x−6)² + 4. x² + 16 = x² − 12x + 36 + 4. 12x = 24. x = 2. The point is (2, 0). Check: both distances are √(4+16)=√20 units.
Check your answer:
2
Find the three points that divide A(−2, 2) and B(4, 8) into four equal parts.
Model answer (board format)
The ratios are 1 : 3, 1 : 1 and 3 : 1. At 1 : 3, x = (1×4 + 3×(−2))/4 = −1/2, y = (1×8 + 3×2)/4 = 7/2. The midpoint is (1, 5). At 3 : 1, x = (3×4 + 1×(−2))/4 = 5/2, y = (3×8 + 1×2)/4 = 13/2. The points are (−1/2, 7/2), (1, 5) and (5/2, 13/2).
Check your answer:
3
One town is at the origin and the other is at (6, 8) kilometres. Find the straight distance. A tower divides the segment in 1 : 3. Write the coordinates of the tower.
Model answer (board format)
Distance √[36 + 64] = √100 = 10 km. Tower: x = (1×6 + 3×0)/4 = 1.5, y = (1×8 + 3×0)/4 = 2. The tower is at (1.5, 2) km. It is nearer the origin, because the ratio is 1 : 3.
Check your answer:
BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
Which of these is a point of the y-axis?
💡 Hint:
Abscissa 0.
Explanation:
(0, −8). (−8, 0) is on the x-axis.
2
The distance between (−2, 1) and (1, 5) is —
💡 Hint:
3² + 4².
Explanation:
5 units.
3
The distance of (8, −6) from the origin is ______ units.
💡 Hint:
10 units.
Explanation:
64 + 36 = 100.
4
Find the distance between (1, −1) and (1, 6) and say what kind of line it is.
Model answer (board format)
Δx = 0, Δy = 7. The distance is 7 units. The x-coordinates match, so the line is parallel to the y-axis.
Check your answer:
5
Find the midpoint M of A(2, −1) and B(8, 5). Then find the distance MA.
Model answer (board format)
M = ((2+8)/2, (−1+5)/2) = (5, 2). MA = √[(5−2)² + (2−(−1))²] = √[9+9] = √18 = 3√2 units. MB is the same length.
Check your answer:
6
Assertion (A): The trisection points of (0, 0) and (9, 0) are (3, 0) and (6, 0).
Reason (R): Trisection uses only the ratio 1 : 1.
Explanation:
A is true. R is false. The ratios are 1 : 2 and 2 : 1. The ratio 1 : 1 is the midpoint.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student divides A(1, 2) and B(7, 8) in 2 : 1 and writes (3, 4) instead of (5, 6). The miss is —
💡 Hint:
2 : 1 means m1 = 2.
Explanation:
The correct point is x = (2×7 + 1×1)/3 = 5, y = (2×8 + 1×2)/3 = 6. (3, 4) is the ratio 1 : 2.
2
On a field, players stand at (0, 0) and (8, 6) metres. The straight distance is —
💡 Hint:
64 + 36.
Explanation:
√100 = 10 m.
3
Assertion (A): (−7, 0) is the same distance from (2, −5) and from (−2, 9).
Reason (R): A point on the x-axis has ordinate 0, and setting the squares of the two distances equal gives x = −7.
Explanation:
Both are true. Both distances are √106 units.
4
Assertion (A): Exercise 7.3 of this book asks for the area of a triangle.
Reason (R): The exercises of this chapter are 7.1 on distance and 7.2 on internal section.
Explanation:
A is false. This book has no 7.3. R is true.
5
A drone flies from (1, 2) to (4, 6). Find the straight distance in metres. Then write the halfway point if it stops at half the distance.
Model answer (board format)
Distance √[9+16] = 5 m. The halfway point is the midpoint ((1+4)/2, (2+6)/2) = (2.5, 4).
Check your answer:
6
The vertices of a parallelogram in order are (1, 2), (4, y), (x, 6), (3, 5). Find x and y. Which fact did you use?
Model answer (board format)
The diagonals meet at one midpoint. (1+x)/2 = (4+3)/2 = 7/2, so x = 6. (2+6)/2 = 4 = (y+5)/2, so y = 3.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
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CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
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🧠 What you learned + equation sheet
What you learned
- The abscissa is x and the ordinate is y. On the x-axis y = 0, and on the y-axis x = 0.
- The distance is √[(x2 − x1)² + (y2 − y1)²]. Take only the positive root.
- The distance from the origin is √(x² + y²).
- If the two smaller distances add up to the largest, the points are collinear.
- In the internal ratio m1 : m2, m1 goes with the second point.
- The midpoint is ((x1 + x2)/2, (y1 + y2)/2). External division is not in this chapter.
| What | Keep this |
|---|---|
| Abscissa / ordinate | (x, y) |
| Distance | √[(x2−x1)²+(y2−y1)²] |
| Origin | √(x²+y²) |
| Collinear | sum of two smaller = largest |
| Section | (m1x2+m2x1)/(m1+m2) |
| Midpoint | ((x1+x2)/2, (y1+y2)/2) |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.