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Title: Baghela
Description: BTp notes have a lot of knowledge which is required by the children at time of study. You can easily learn your course or remained at time of exam.
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NCERT Solutions for Class 10 Maths Unit 7
Coordinate Geometry Class 10
Unit 7 Coordinate Geometry Exercise 7
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2, 7
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4 Solutions
Exercise 7
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Can you now find the distance between the two
towns A and B discussed in Section 7
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Answer :
Distance between points
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Yes, we can find the distance between the given towns A and B
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Therefore, town B will be at point (36, 15) with respect to town A
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Q3 :
Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear
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Let
Therefore, the points (1, 5), (2, 3), and ( - 2, - 11) are not collinear
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Answer :
Let the points (5, - 2), (6, 4), and (7, - 2) are representing the vertices A, B, and C of the given triangle respectively
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Q5 :
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure
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Using distance formula, find which of them is correct
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It can be observed that all sides of this quadrilateral ABCD are of the same length and also the diagonals are of the
same length
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It can be observed that all sides of this quadrilateral are of the same length and also, the diagonals are of the same
length
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(ii)Let the points ( - 3, 5), (3, 1), (0, 3), and ( - 1, - 4) be representing the vertices A, B, C, and D of the given
quadrilateral respectively
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Therefore, it can be said that it is only a
general quadrilateral, and not specific such as square, rectangle, etc
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It can be observed that opposite sides of this quadrilateral are of the same length
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Therefore, the given points are the vertices of a parallelogram
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Answer :
We have to find a point on x-axis
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Let the point on x-axis be
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Therefore, the point is ( - 7, 0)
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Answer :
It is given that the distance between (2, - 3) and (10, y) is 10
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Also find the distance QR and PR
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When point R is (4, 6),
When point R is ( - 4, 6),
Q10 :
If Q (0, 1) is equidistant from P (5, - 3) and R (x, 6), find the values of x
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Answer :
Therefore, point R is (4, 6) or ( - 4, 6)
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Answer :
Point (x, y) is equidistant from (3, 6) and ( - 3, 4)
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Answer :
Point (x, y) is equidistant from (3, 6) and ( - 3, 4)
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2 : Solutions of Questions on Page Number : 167
Q1 :
Find the coordinates of the point which divides the join of (- 1, 7) and (4, - 3) in the ratio 2:3
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Using the section formula, we obtain
Therefore, the point is (1, 3)
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Answer :
Let P(x, y) be the required point
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Q3 :
Find the coordinates of the points of trisection of the line segment joining (4, - 1) and (- 2, - 3)
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Point Q divides AB internally in the ratio 2:1
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Answer :
Let P (x1, y1) and Q (x2, y2) are the points of trisection of the line segment joining the given points i
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, AP = PQ = QB
Therefore, point P divides AB internally in the ratio 1:2
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Q5 :
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn
with chalk powder at a distance of 1 m each
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Niharika runs
the distance AD on the 2nd line and
posts a green flag
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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?
Answer :
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It can be observed that Niharika posted the green flag at
of the distance AD i
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,
starting point of 2nd line
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Similarly, Preet posted red flag at
of the distance AD i
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,
Therefore, the coordinates of this point R are (8, 20)
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Distance between these flags by using distance formula = GR
=
The point at which Rashmi should post her blue flag is the mid-point of the line joining these points
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Therefore, Rashmi should post her blue flag at 22
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Q6 :
To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn
with chalk powder at a distance of 1 m each
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Niharika runs
the distance AD on the 2nd line and
posts a green flag
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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?
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Answer :
It can be observed that Niharika posted the green flag at
of the distance AD i
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,
nd
starting point of 2 line
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Similarly, Preet posted red flag at
of the distance AD i
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,
Therefore, the coordinates of this point R are (8, 20)
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Distance between these flags by using distance formula = GR
=
The point at which Rashmi should post her blue flag is the mid-point of the line joining these points
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Therefore, Rashmi should post her blue flag at 22
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Q7 :
Find the ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6)
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Q8 :
Find the ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6)
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Q9 :
Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis
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Answer :
Let the ratio in which the line segment joining A (1, - 5) and B ( - 4, 5) is divided by x-axisbe
Therefore, the coordinates of the point of division is
We know that y-coordinate of any point on x-axis is 0
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Division point =
Q10 :
Find the ratio in which the line segment joining A (1, - 5) and B (- 4, 5) is divided by the x-axis
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Answer :
Let the ratio in which the line segment joining A (1, - 5) and B ( - 4, 5) is divided by x-axisbe
Therefore, the coordinates of the point of division is
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We know that y-coordinate of any point on x-axis is 0
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Division point =
Q11 :
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y
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Intersection
point O of diagonal AC and BD also divides these diagonals
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If O is the mid-point of AC, then the coordinates of O are
If O is the mid-point of BD, then the coordinates of O are
Since both the coordinates are of the same point O,
Q12 :
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y
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Intersection
point O of diagonal AC and BD also divides these diagonals
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If O is the mid-point of AC, then the coordinates of O are
If O is the mid-point of BD, then the coordinates of O are
Since both the coordinates are of the same point O,
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Q13 :
Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, - 3) and B is (1, 4)
Answer :
Let the coordinates of point A be (x, y)
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Q14 :
Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, - 3) and B is (1, 4)
Answer :
Let the coordinates of point A be (x, y)
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Q15 :
If A and B are ( - 2, - 2) and (2, - 4), respectively, find the coordinates of P such that
on the line segment AB
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Since
,
Therefore, AP: PB = 3:4
Point P divides the line segment AB in the ratio 3:4
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Answer :
The coordinates of point A and B are ( - 2, - 2) and (2, - 4) respectively
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Q17 :
Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal
parts
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Q18 :
Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal
parts
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Q19 :
Find the area of a rhombus if its vertices are (3, 0), (4, 5), ( - 1, 4) and ( - 2, - 1) taken in order
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Q20 :
Find the area of a rhombus if its vertices are (3, 0), (4, 5), ( - 1, 4) and ( - 2, - 1) taken in order
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Exercise 7
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(i) (7, - 2), (5, 1), (3, - k) (ii) (8, 1), (k, - 4), (2, - 5)
Answer :
(i) For collinear points, area of triangle formed by them is zero
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Therefore, for points (8, 1), (k, - 4), and (2, - 5), area = 0
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Q4 :
In each of the following find the value of 'k', for which the points are collinear
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Therefore, for points (7, - 2) (5, 1), and (3, k), area = 0
(ii) For collinear points, area of triangle formed by them is zero
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Find the ratio of this area to the area of the given triangle
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Let D, E, F be the mid-points of the sides of this triangle
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Find the ratio of this area to the area of the given triangle
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Let D, E, F be the mid-points of the sides of this triangle
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Join AC to form two triangles
ΔABC and ΔACD
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Join AC to form two triangles
ΔABC and ΔACD
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Verify this
result for ΔABC whose vertices are A (4, - 6), B (3, - 2) and C (5, 2)
Answer :
Let the vertices of the triangle be A (4, - 6), B (3, - 2), and C (5, 2)
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Therefore, AD is the median in ΔABC
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Therefore, area of ΔABD is 3 square units
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Therefore, area of ΔADC is 3 square units
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Q10 :
You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas
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Let D be the mid-point of side BC of ΔABC
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However, area cannot be negative
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However, area cannot be negative
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Clearly, median AD has divided ΔABC in two triangles of equal areas
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4 : Solutions of Questions on Page Number : 171
Q1 :
Determine the ratio in which the line 2x + y - 4 = 0 divides the line segment joining the points A(2, - 2) and
B(3, 7)
Answer :
Let the given line divide the line segment joining the points A(2, - 2) and B(3, 7) in a ratio k : 1
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Q2 :
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Determine the ratio in which the line 2x + y - 4 = 0 divides the line segment joining the points A(2, - 2) and
B(3, 7)
Answer :
Let the given line divide the line segment joining the points A(2, - 2) and B(3, 7) in a ratio k : 1
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Q3 :
Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear
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This is the required relation between x and y
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Answer :
If the given points are collinear, then the area of triangle formed by these points will be 0
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Q5 :
Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3)
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And let the points (6, - 6), (3, - 7), and (3, 3) be representing the points A, B,
and C on the circumference of the circle
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Q6 :
Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3)
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And let the points (6, - 6), (3, - 7), and (3, 3) be representing the points A, B,
and C on the circumference of the circle
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Q7 :
The two opposite vertices of a square are (- 1, 2) and (3, 2)
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Answer :
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Let ABCD be a square having ( - 1, 2) and (3, 2) as vertices A and C respectively
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We know that the sides of a square are equal to each other
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In ΔABC,
AB2 + BC2 = AC2
⇒ 4 + y2 + 4 - 4y + 4 + y2 - 4y + 4 =16
⇒ 2y2 + 16 - 8 y =16
⇒ 2y2 - 8 y = 0
⇒ y (y - 4) = 0
⇒ y = 0 or 4
We know that in a square, the diagonals are of equal length and bisect each other at 90°
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Therefore, it will also be the mid-point of BD
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Q8 :
The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for
their gardening activity
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There is a triangular grassy lawn in the plot as shown in the following figure
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(i) Taking A as origin, find the coordinates of the vertices of the triangle
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What do you observe?
Answer :
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(i) Taking A as origin, we will take AD as x-axis and AB as y-axis
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(ii) Taking C as origin, CB as x-axis, and CD as y-axis, the coordinates of vertices P, Q, and R are (12, 2), (13, 6),
and (10, 3) respectively
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Q9 :
The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for
their gardening activity
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There is a triangular grassy lawn in the plot as shown in the following figure
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(i) Taking A as origin, find the coordinates of the vertices of the triangle
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What do you observe?
Answer :
(i) Taking A as origin, we will take AD as x-axis and AB as y-axis
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(ii) Taking C as origin, CB as x-axis, and CD as y-axis, the coordinates of vertices P, Q, and R are (12, 2), (13, 6),
and (10, 3) respectively
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Q10 :
The vertices of a ΔABC are A (4, 6), B (1, 5) and C (7, 2)
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Calculate the area of the ΔADE and compare it with the area of
ΔABC
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6 related to
ratio of areas of two similar triangles)
Answer :
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Given that,
Therefore, D and E are two points on side AB and AC respectively such that they divide side AB and AC in a ratio of
1:3
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Alternatively,
We know that if a line segment in a triangle divides its two sides in the same ratio, then the line segment is parallel to
the third side of the triangle
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Hence, the ratio between the areas of these two triangles will be the square of the ratio between the sides of these
two triangles
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A line is drawn to intersect sides AB and AC at D and
E respectively, such that
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(Recall Converse of basic proportionality theorem and Theorem 6
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Clearly, the ratio between the areas of ΔADE and ΔABC is 1:16
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These two triangles so formed (here ΔADE and ΔABC) will be similar to each other
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Therefore, ratio between the areas of ΔADE and ΔABC =
Q12 :
Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ΔABC
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Find the coordinates of point D
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(iv) What do you observe?
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(v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ΔABC, find the coordinates of the centroid of the
triangle
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Therefore, D is the mid-point of side BC
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(iii) Median BE of the triangle will divide the side AC in two equal parts
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Point Q divides the side BE in a ratio 2:1
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Therefore, F is the mid-point of side AB
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(iv) It can be observed that the coordinates of point P, Q, R are the same
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(v) Consider a triangle, ΔABC, having its vertices as A(x1, y1), B(x2, y2), and C(x3,
y3)
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Therefore, D is the mid-point of side BC
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Point O divides the side AD in a ratio 2:1
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(i) The median from A meets BC at D
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(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1
(iii) Find the coordinates of point Q and R on medians BE and CF respectively such that BQ: QE = 2:1 and
CR: RF = 2:1
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Answer :
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(i) Median AD of the triangle will divide the side BC in two equal parts
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(ii) Point P divides the side AD in a ratio 2:1
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Therefore, E is the mid-point of side AC
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Median CF of the triangle will divide the side AB in two equal parts
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Point R divides the side CF in a ratio 2:1
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Therefore, all these are representing the same point on the plane i
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, the centroid of the triangle
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Median AD of the triangle will divide the side BC in two equal parts
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Let the centroid of this triangle be O
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Q14 :
ABCD is a rectangle formed by the points A (- 1, - 1), B (- 1, 4), C (5, 4) and D (5, - 1)
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Is the quadrilateral PQRS is a square? a rectangle? or a
rhombus? Justify your answer
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However, the diagonals are of
different lengths
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Q15 :
ABCD is a rectangle formed by the points A (- 1, - 1), B (- 1, 4), C (5, 4) and D (5, - 1)
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Is the quadrilateral PQRS is a square? a rectangle? or a
rhombus? Justify your answer
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However, the diagonals are of
different lengths
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Title: Baghela
Description: BTp notes have a lot of knowledge which is required by the children at time of study. You can easily learn your course or remained at time of exam.
Description: BTp notes have a lot of knowledge which is required by the children at time of study. You can easily learn your course or remained at time of exam.