Exercise 20.3
Page no 20.26
Type 1
Question 1
Find the coordinates of the point which divides the line segment joining 24 and $(6,8)$ in the ratio $1: 3$ internally and externally.
Question 2
Find the coordinates of the points which trisect the line segment joining $t=$ points $(2,3)$ and $(6,5)$.
Question 3
$A(1,4)$ and $B(4,8)$ are two points. $P$ is a point on $A B$ such that $A P=A B+BP$ If $A P=10$, find the coordinates of $P$.
Question 4
The line segment joining $A(2,3)$ and $B(-3,5)$ is extended through each ed tr a length equal to its original length. Find the coordinates of the new Ends
Question 5
The line segment joining $A(6,3)$ to $B(-1,-4)$ is doubled in length by having half its length added to each end. Find the coordinates of the new ends.
Question 6
The coordinates of two points $A$ and $B$ are $(-1,4)$ and $(5,1)$, respectively. Find the coordinates of the point $P$ which lie on extended line $A B$ such that 11 three times as far from $B$ as from $A$.
Question 7
Find the distance of that point from the origin which divides the line segment joining the points $(5,-4)$ and $(3,-2)$ in the ratio $4: 3$.
Question 8
The coordinate of the middle points of the sides of a triangle are (1,1), (2,3) and (4,1) Find the coordinates of its vertices
Question 9
A(1.-2) and B(2,5) are two points . The lines OA, OB are produced to C and D respectively such that OC= 2OA and OD = 2 OB. Find CD
Question 10
Find the length of the medians of the triangle whose vertices $(-1,3),(1,-1)$ and $(5,1)$.
Question 11
If $A(1,5), B(-2,1)$ and $C(4,1)$ be the vertices of $\triangle A B C$ and internal bisects ${ }^{n}$ $\angle A$ meets $B C$ at $D$, find $A D$.
Page no 20.27
Question 12
If the middle point of the line segment joining $(3,4)$ and $(k, 7)$ is (x,y) and 2x+ 2Y + 1= 0. find the value of k.
Question 13
One end of a diameter of a circle is at (2,3) and the center is (2,-5) Find the Coordinate of the other end of the diameter
Question 14
If the point C(-1,2) Divides internally the line segment joining A (2,5) and B in the ratio3: 4 Find the coordinates of B
Question 15
Find the ratio does the x-axis divides the line segment joining the points $(2,-2)$ and $(-4,1)$
Question 16
In what ratio does the x-axis divides the line segment joining $(2-3)$ and $(5,6)$
Question 17
Find the ratio in which the join of the points $(1,2)$ and $(-2,3)$ is divided by the line $3 x+4 y=7$
Type 2
Question 18
Find the centroid and incentre of the triangle whose vertices are $(2,4),(6,4),(2,0)$
Question 19
The vertices of a triangle are at $(2,2),(0,6)$ and $(8,10)$. Find the coordinates of the trisection point of each median which is nearer the opposite side.
Question 20
Two vertices of a triangle are $(1,4)$ and $(5,2)$. If its centroid is $(0,-3)$, find the third vertex.
Question 21
The coordinates of centroid of a triangle are $(\sqrt{3}, 2)$ and two of its vertices are $(2 \sqrt{3},-1)$ and $(2 \sqrt{3}, 5)$. Find the third vertex of the triangle.
Question 22
Find the centroid of the triangle $A B C$ whose vertices are $A(9,2), B(1,10)$ and $C(-7,-6)$. Find the coordinates of the middle points of its sides and hence find the centroid of the triangle formed by joining these middle points. Do the the triangles have same centroid?
Question 23
If $(1,2),(0,-1)$ and $(2,-1)$ are the middle points of the sides of a triangle, find the coordinates of its centroid.
Question 24
Find the incentre of the triangle with vertices $(1, \sqrt{3}),(0,0)$ and $(2,0)$.
[Hint : Given triangle is equilateral therefore its incentre and centroid will be same point.]
Question 25
The mid-points of the sides of a triangles are $\left(\frac{1}{2}, 0\right),\left(\frac{1}{2}, \frac{1}{2}\right)$ and $\left(0, \frac{1}{2}\right)$. Find the coordinates of the incentre.
Question 26
Two vertices of a triangle are $A(2,1)$ and $B(3,-2)$. The third vertex $C$ lies on the line $y=x+9$. If the centroid of $\triangle A B C$ lies on $y$-axis, find the coordinates of $\mathrm{C}$ and the centroid.
Question 27
Prove that the points $(-2,-1),(1,0),(4,3)$ and $(1,2)$ are the vertices of a Parallelogram.
Question 28
Show that the points $A(1,0), B(5,3), C(2,7)$ and $D(-2,4)$ are the vertices of a rhombus.
Question 29
Prove that the points $(4,8),(0,2),(3,0)$ and $(7,6)$ are the vertices of a rectangle
Question 30
Prove that the points $(4,3),(6,4),(5,6)$ and $(3,5)$ are the vertices of a square.
Question 31
If (6,8) (3,7) and (-2,-2) be the coordinates of the three Consecutive vertices of a parallelogram . Find the coordinates of the fourth vertex
Page no 20.28
Question 32
Three consecutive vertices of a rhombus are $(5,3),(2,7)$ and $(-2,4)$ Find the fourth vertex
Question 33
A quadrilateral has the vertices at the points $(-4,2),(2,6),(8,5)$ and (9,-7)Show that the mid-points of the sides of this quadrilateral are the vertices of a parallelogram
Question 34
Prove that the line segment joining the middle points of two sides of a triangle is half the third side.
Question 35
If $P, Q, R$ divide the sides $B C, C A$ and $A B$ of $\triangle A B C$ in the same ratio, prove that the centroid of the triangles $A B C$ and $P Q R$ coincide.
Type 3
Question 35
Prove that in any triangle four times the sum of the squares of the median is equal to three times the sum of the squares of the sides.
Question 36
If $G$ be the centroid of $\triangle A B C$, prove that
$A B^{2}+B C^{2}+C A^{2}=3\left(G A^{2}+G B^{2}+G C^{2}\right)$
Question 37
Show that the middle point of the hypotenuse of a right angled triangle in equidistant from its vertices.
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