Class 10 Mathematics • Coordinate Geometry • Chapter 7
Exam Focus:
These questions cover the most important
concepts and applications from Coordinate Geometry.
First solve the question yourself.
Click
🇮🇳 हिंदी में समझें
for an easy Hindi explanation.
Click
🇮🇳 हिंदी में Solution देखें
to see the complete Hindi solution.
Important Topics:
Distance Formula, Section Formula,
Midpoint, Trisection, Equidistant Points,
Missing Coordinates and Geometrical Applications.
A. Distance Formula
Important Question 1
Distance Between Two Points
Find the distance between the points
A(3, 4) and
B(7, 1).
हिंदी में समझें:
Distance Formula का उपयोग करें:
दोनों points के x-coordinates का difference
और y-coordinates का difference निकालकर
उनके squares करें।
फिर दोनों को जोड़कर square root लें।
Step-by-Step Solution:
Distance AB
The coordinate differences form the two perpendicular
sides of a right triangle.
Expanding and simplifying:
−14x − 2y + 50
=
−6x − 10y + 34
−8x + 8y + 16 = 0
−x + y + 2 = 0
x − y = 2
✅ Final Answer:
x − y = 2
B. Section Formula
Important Question 5
Section Formula
Find the coordinates of the point P which divides
the line segment joining A(2,3) and B(8,9)
internally in the ratio 2:1.
हिंदी में समझें:
Section Formula में ratio m:n दिया गया है।
यहाँ m = 2 और n = 1.
Formula में B के coordinates को m से
और A के coordinates को n से multiply करें।
Step-by-Step Solution:
Internal Division
P =
(
2×8 + 1×2
2+1
,
2×9 + 1×3
2+1
)
= (6,7)
✅ Final Answer:
P = (6,7)
Important Question 6
Midpoint Formula
Find the midpoint of the line segment joining
A(−4,2) and B(6,8).
हिंदी में समझें:
Midpoint में दोनों x-coordinates का average
और दोनों y-coordinates का average लेते हैं।
Step-by-Step Solution:
Midpoint M
M =
(
−4 + 6
2
,
2 + 8
2
)
= (1,5)
✅ Final Answer:
M = (1,5)
Important Question 7
Trisection of a Line Segment
Find the coordinates of the two points which trisect
the line segment joining A(2,−2) and B(−7,4).
हिंदी में समझें:
Trisection का मतलब segment को 3 equal parts
में divide करना है।
पहला point ratio 1:2 और दूसरा point ratio 2:1
में मिलेगा।
Step-by-Step Solution:
Two Trisection Points
First point P:
P divides AB in ratio 1:2.
P =
(
1(−7) + 2(2)
3
,
1(4) + 2(−2)
3
)
= (−1,0)
Second point Q:
Q divides AB in ratio 2:1.
Q =
(
2(−7) + 1(2)
3
,
2(4) + 1(−2)
3
)
= (−4,2)
✅ Final Answer:
P = (−1,0), Q = (−4,2)
Important Question 8
Ratio from a Given Point
Find the ratio in which the point P(5,6)
divides the line segment joining A(1,2) and B(7,8).
हिंदी में समझें:
मान लें ratio m:n है।
Section Formula में P के x और y coordinates
के लिए equations बनाकर m:n निकालें।
Step-by-Step Solution:
5 =
7m + n
m+n
5m + 5n = 7m + n
4n = 2m
m:n = 2:4 = 1:2
✅ Final Answer:
AP : PB = 1 : 2
C. Geometrical Applications
Important Question 9
Isosceles Triangle
Show that the points A(1,2), B(4,6) and C(7,2)
form an isosceles triangle.
हिंदी में समझें:
तीनों sides की lengths Distance Formula से निकालें।
अगर दो sides बराबर मिलती हैं तो triangle
isosceles होगा।
Step-by-Step Solution:
Triangle ABC
AB² = (4−1)² + (6−2)²
= 9 + 16 = 25
AB = 5
BC² = (7−4)² + (2−6)²
= 9 + 16 = 25
BC = 5
Since AB = BC, the triangle is isosceles.
✅ Final Answer:
AB = BC = 5 units
Therefore ΔABC is isosceles.
Important Question 10
Equilateral Triangle Check
Determine whether the points
A(0,0), B(4,0) and C(2,2√3)
form an equilateral triangle.
हिंदी में समझें:
Equilateral triangle में सभी तीन sides equal होती हैं।
इसलिए AB, BC और CA तीनों distances निकालें।
Step-by-Step Solution:
Equilateral Triangle
AB = 4
BC =
√[(2−4)² + (2√3−0)²]
= √[4 + 12]
= 4
CA =
√[(2−0)² + (2√3−0)²]
= √[4+12]
= 4
✅ Final Answer:
AB = BC = CA = 4 units.
Therefore ΔABC is equilateral.
Important Question 11
Collinearity Using Distance
Show that the points
A(1,2), B(3,4) and C(5,6)
are collinear.
हिंदी में समझें:
अगर AB + BC = AC हो, तो तीनों points
एक ही straight line पर होंगे।
Step-by-Step Solution:
AB = √[(3−1)²+(4−2)²]
= √8
BC = √[(5−3)²+(6−4)²]
= √8
AC = √[(5−1)²+(6−2)²]
= √32
= 2√8
Since
AB + BC = √8 + √8 = 2√8 = AC,
the three points are collinear.
✅ Final Answer:
A, B and C are collinear.
Important Question 12
Rectangle Verification
Show that the points A(1,1), B(6,1), C(6,4)
and D(1,4) form a rectangle.
हिंदी में समझें:
Opposite sides की lengths compare करें और
diagonals की lengths भी check करें।
Step-by-Step Solution:
Quadrilateral ABCD
AB = √[(6−1)² + (1−1)²] = 5
BC = √[(6−6)² + (4−1)²] = 3
CD = 5
DA = 3
AC = √[(6−1)² + (4−1)²]
= √34
BD = √[(1−6)² + (4−1)²]
= √34
Opposite sides are equal and both diagonals are equal.
Therefore the quadrilateral is a rectangle.
✅ Final Answer:
ABCD is a rectangle.
Important Question 13
Find the Ratio
Point P(3,4) divides the line segment joining
A(1,2) and B(7,8). Find the ratio AP:PB.
हिंदी में समझें:
Section Formula में ratio को m:n मान लें।
P के x-coordinate से equation बनाकर ratio निकालें।
Step-by-Step Solution:
3 =
7m + n
m+n
3m + 3n = 7m + n
2n = 4m
n = 2m
m:n = 1:2
✅ Final Answer:
AP : PB = 1 : 2
Important Question 14
Internal Division with Fractions
Find the coordinates of the point P that divides
A(−2,3) and B(4,−5) internally in the ratio 3:2.
हिंदी में समझें:
m = 3 और n = 2 लें।
Section Formula में numerator और denominator
को carefully substitute करें।
Step-by-Step Solution:
P =
(
3(4) + 2(−2)
3+2
,
3(−5) + 2(3)
3+2
)
=
(
12−4
5
,
−15+6
5
)
=
(8/5, −9/5)
✅ Final Answer:
P = (8/5, −9/5)
Important Question 15
Midpoint and Distance
The midpoint of A(2,3) and B(x,y) is
(5,7). Find x and y.
हिंदी में समझें:
Midpoint Formula में दिए गए midpoint coordinates
substitute करें और x तथा y की values निकालें।
Step-by-Step Solution:
5 =
2+x
2
10 = 2+x
x = 8
7 =
3+y
2
14 = 3+y
y = 11
✅ Final Answer:
x = 8, y = 11
Important Question 16
Distance Formula Reasoning
Find the value of k if the distance between
P(k,3) and Q(4,7) is 5 units.
हिंदी में समझें:
Distance Formula लगाकर square both sides करें।
फिर quadratic equation solve करें।
Find the coordinates of the point dividing the line
segment joining A(−5,7) and B(4,−2) in the ratio 1:2.
हिंदी में समझें:
Ratio 1:2 का मतलब point A से ज्यादा पास होगा।
Section Formula directly use करें।
Step-by-Step Solution:
P =
(
1(4)+2(−5)
3
,
1(−2)+2(7)
3
)
= (−2,4)
✅ Final Answer:
P = (−2,4)
Important Question 20
Coordinate Geometry Board Problem
The points A(−2,3), B(4,5) and C(10,7) are given.
Show that B is the midpoint of AC.
हिंदी में समझें:
AC का midpoint निकालें। अगर midpoint exactly
B(4,5) आता है तो B, AC का midpoint है।
Step-by-Step Solution:
B as Midpoint of AC
Midpoint of AC =
(
−2 + 10
2
,
3 + 7
2
)
= (4,5)
Since the midpoint is B(4,5), B is the midpoint of AC.
✅ Final Answer:
B is the midpoint of AC.
⭐ Quick Exam Revision
Before solving a Coordinate Geometry question:
1. Write the coordinates clearly.
2. Identify whether the question asks for
distance, midpoint or division in a ratio.
3. Write the formula first.
4. Substitute values carefully.
5. Keep brackets around negative numbers.
6. Simplify the final answer completely.