10th Maths 7.2
NCERT Class 10th solution of Exercise 7.1
NCERT Class 10th Maths Projects
Exercise 7.2
Q1. Find the coordinates of the point which divides the join of (-1,7) and (4,-3) in the ratio 2:3.
Sol. :
Given:
Let x1=-1,y1=7,x2=4,y2=3 and m1=2,m2=3
To find:
The coordinates of the point.
Solve:
Section formula
x=m1x2+m2x1m1+m2
x=2×4+3×(-1)2+3
x=8-35
x=55
x=1
y=m1y2+m2y1m1+m2
y=2×(-3)+3×72+3
y=-6+215
y=155
y=3
Answer:
The coordinates of the point is (1,3).
Q2. Find the coordinates of the points of trisection of the line segment joining (4,-1) and (-2,-3).
Sol. :
Given:
Let x1=4,y1=-1,x2=-2,y2=-3.
To Find:
The coordinates of the points.
Solve:
For P
m1=1,m2=2
x′=m1x2+m2x1m1+m2
x′=1×(-2)+2×41+2
x′=-2+83
x′=63
x′=2
y′=m1y2+m2y1m1+m2
y′=1×(-3)+2×(-1)1+2
y′=-3-23
y=-53
For Q
m1=2,m2=1
x′′=m1x2+m2x1m1+m2
x′′=2×(-2)+1×42+1
x′′=-4+43
x′′=03
x′′=0
y′′=m1y2+m2y1m1+m2
y′′=2×(-3)+1×(-1)2+1
y′′=-6-13
y′′=-73
Answer:
The coordinates of the points are (2,-53) and (0,-73).
Q3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in figure. Niharika runs 14 th the distance AD on the 2nd line and posts a green flag. Preet runs 15th the distance AD on the eight line and post a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?
Sol. :
Let (2,1004)=(2,25),(8,1005)=(8,20),(x,y)
To Find:
The distance between both flags, and the distance of the blue flag post.
Solve:
=√(8-2)2+(20-25)2
=√(6)2+(-5)2
=√36+25
=√61m
x=8+22=102=5m
y=20+252=452=22.5m
Answer:
The distance between both flags is √61m and the distance of the blue flag post is 22,5m.
Q4. Find the ratio in which the line segment joining the points (-3,10) and (6,-8) is divided by (-1,6).
Sol. :
Given:
Let x=-1,y=6 and x1=-3,y1=10,x2=6,y2=-8
To Find :
m1 and m2
Solve:
x=m1x2+m2x1m1+m2
-1=m2×6+m2×(-3)m1+m2
-1=6m1-3m2m1+m2
-m1-m2=6m1-3m2
6m1+m1=3m2-m2
7m1=2m2
m1m2=27
m1:m2=2:7
Answer:
The m1=2,m2=7.
Q5. Find the ratio in which the line segment joining A(1,-5) and B(-4,5) is divided by the x-axis. Also find the coordinates of the point of division.
Sol. :
Given:
Let P(x,0) of x-axis, A(1,-5),B(-4,5)
To Find:
The ratio and the coordinates of the point of division.
Solve:
y=m1y2+m2y1m1+m2
0=m1×5+m2×(-5)m1+m2
0=5m1-5m2
m1=m2
m1:m2=1:1
P is mid-point of AB
x=x1+x21+1=1-42=-32
P(-32,0)
Answer:
The ratio is 1:1 and the coordinates of the point of division (-32,0).
Q6. If (1,2),(4,y),(x,6) and (3,5) are the vertices of a parallelogram taken in order, find x and y.
Sol. :
Let the vertices of parallelogram A(1,2),B(4,y),C(x,6) and (3,5)
To Find:
The value of x and y.
Solve:
The diagonals AC and BD of the parallelogram bisect each other at O(x′,y′)
midpoint of AC = midpoint of BD
(1+x2,2+62)=(4+32,y+52)
(1+x2,82)=(72,y+52)
(1+x2,4)=(72,y+52)
1+x2=72
1+x=7
x=7-1
x=6
and
y+52=4
y+5=8
y=8-5
y=3
Answer:
The value of x=6 and y=3.
Q7. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2,-3) and B is (1,4).
Sol. :
The Centre of the circle is O(2,-3) and point B(1,4).
To Find:
The coordinates of a point A.
Solve:
The centre O of the circle is the midpoint of the diameter AB.
O=A+B2
(2,3)=(x+12,y+42)
2=x+12
4=x+1
x=3
and
y=y+42
2y=y+4
2y-y=4
y=4
Answer:
The coordinates of a point A(3,4).
Q8. If A and B are (-2,-2) and (2,-4), respectively, find the coordinates of P such that AP=37AB and P lies on the line segment AB.
Sol:
Given:
A(-2,-2),B(2,-4) and AP=37AB
To Find:
The coordinates of P.
Solve:
`BP=AB-AP
BP=AB-37AB [AP=37AB]
BP=7AB-3AB7
BP=47AB
Point P divides the segment AB in 3:4.
P(x,y)=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)
x=3×2+4×(-2)3+4=6-87=-27
y=3×(-4)+4×(-2)3+4=-12-87=-207
Answer:
The coordinates of P(-27,-207).
Q9. Find the coordinates of the points which divide the line segment joining A(-2,2) and B(2,8) into four equal parts.
Sol. :
A(-2,2),B(2,8)
To Find:
The coordinates of the points divide AB into four equal points.
Solve:
Q is the midpoint of AB
Q(x′′,y′′)=(-2+22,2+82)=(0,5)
P is the midpoint of AQ
P(x′,y′)=(-2+02,2+52)=(-1,72)
R is the midpoint of QB
R(x′′′,y′′′)=(0+22,5+82)=(1,132)
Answer:
The coordinates of the points divide AB into four equal points are Q(0,5),P(-1,72),R(1,132).
Q10. Find the area of a rhombus if its vertices are (3,0),(4,5),(-1,4) and (-2,-1) takes in order. [Hint: Area of a rhombus=12(product of its diagonals)]
Sol. :
Given:
Let vertices of rhombus are A(3,0),B(4,5),C(-1,4) and D(-2,-1).
To Find:
Area of a rhombus.
Solve:
Diagonal AC
AC=√(-1-3)2+(4-0)2
AC=√(-4)2+(4)2
AC=√16+16=4√2
Diagonal BD
BD=√(-2-4)2+(-1-5)2
BD=√(-6)2+(-6)2
BD=√36+36
BD=6√2
Area of a rhombus=12(AC×BD)
=12(4√2×6√2)
=24 square units
Answer:
The area of a rhombus 24 square units.
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