ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11

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5 years ago

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11

Question 1.
Find the co-ordinates of the mid-point of the line segments joining the following pairs of points:
(i) (2, -3), ( -6, 7)
(ii) (5, -11), (4, 3)
(iii) (a + 3, 5b), (2a – 1, 3b + 4)
Solution:

Question 2.
The co-ordinates of two points A and B are (-3, 3) and (12, -7) respectively. P is a point on the line segment AB such that AP : PB = 2 : 3. Find the co-ordinates of P.
Solution:

Question 3.
P divides the distance between A (-2, 1) and B (1, 4) in the ratio of 2 : 1. Calculate the co-ordinates of the point P.
Solution:

Question 4.
(i) Find the co-ordinates of the points of trisection of the line segment joining the point (3, -3) and (6, 9).
(ii) The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, – 2) and \(\left( \frac { 5 }{ 3 } ,q \right) \) respectively, find the values of p and q.
Solution:

Question 5.
(i) The line segment joining the points A (3, 2) and B (5, 1) is divided at the point P in the ratio 1 : 2 and it lies on the line 3x – 18y + k = 0. Find the value of k.
(ii) A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that \(\frac { AP }{ PB } =\frac { k }{ 1 } \) If P lies on the line x + y = 0, then find the value of k.
Solution:

Question 6.
Find the coordinates of the point which is three-fourths of the way from A (3, 1) to B (-2, 5).
Solution:

Question 7.
Point P (3, -5) is reflected in P’ in the x-axis. Also, P on reflection in the y-axis is mapped as P”.
(i) Find the co-ordinates of P’ and P”.
(ii) Compute the distance P’ P”.
(iii) Find the middle point of the line segment P’ P”.
(iv) On which co-ordinate axis does the middle point of the line segment P P” lie?
Solution:

Question 8.
Use graph paper for this question. Take 1 cm = 1 unit on both axes. Plot the points A(3, 0) and B(0, 4).
(i) Write down the co-ordinates of A1, the reflection of A in the y-axis.
(ii) Write down the co-ordinates of B1, the reflection of B in the x-axis.
(iii) Assign the special name to the quadrilateral ABA1B1.
(iv) If C is the midpoint is AB. Write down the co-ordinates of the point C1, the reflection of C in the origin.
(v) Assign the special name to quadrilateral ABC1B1.
Solution:

Question 9.
The line segment joining A (-3, 1) and B (5, -4) is a diameter of a circle whose centre is C. find the co-ordinates of the point C. (1990)
Solution:

Question 10.
The mid-point of the line segment joining the points (3m, 6) and (-4, 3n) is (1, 2m – 1). Find the values of m and n.
Solution:

Question 11.
The co-ordinates of the mid-point of the line segment PQ are (1, -2). The co-ordinates of P are (-3, 2). Find the co-ordinates of Q.(1992)
Solution:

Question 12.
AB is a diameter of a circle with centre C (-2, 5). If point A is (3, -7). Find:
(i) the length of radius AC.
(ii) the coordinates of B.
Solution:

Question 13.
Find the reflection (image) of the point (5, -3) in the point (-1, 3).
Solution:

Question 14.
The line segment joining A \(\left( -1,\frac { 5 }{ 3 } \right) \) the points B (a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects y-axis. Calculate
(i) the value of a
(ii) the co-ordinates of P. (1994)
Solution:

Question 15.
The point P (-4, 1) divides the line segment joining the points A (2, -2) and B in the ratio of 3 : 5. Find the point B.
Solution:

Question 16.
(i) In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7 ,6) ?
(ii) In what ratio does the point (-4, b) divide the line segment joining the points P (2, -2), Q (-14, 6) ? Hence find the value of b.
Solution:

Question 17.
The line segment joining A (2, 3) and B (6, -5) is intercepted by the x-axis at the point K. Write the ordinate of the point k. Hence, find the ratio in which K divides AB. Also, find the coordinates of the point K.
Solution:

Question 18.
If A (-4, 3) and B (8, -6)
(i) find the length of AB.
(ii) in what ratio is the line joining AB, divided by the x-axis? (2008)
Solution:

Question 19.
(i) Calculate the ratio in which the line segment joining (3, 4) and(-2, 1) is divided by the y-axis.
(ii) In what ratio does the line x – y – 2 = 0 divide the line segment joining the points (3, -1) and (8, 9)?
Also, find the coordinates of the point of division.
Solution:

Question 20.
Given a line segment AB joining the points A (-4, 6) and B (8, -3). Find:
(i) the ratio in which AB is divided by the y-axis.
(ii) find the coordinates of the point of intersection.
(iii)the length of AB.
Solution:

Question 21.
(i) Write down the co-ordinates of the point P that divides the line joining A (-4, 1) and B (17, 10) in ratio 1 : 2.
(ii)Calculate the distance OP where O is the origin.
(iii)In what ratio does the y-axis divide the line AB?
Solution:

Question 22.
Calculate the length of the median through the vertex A of the triangle ABC with vertices A (7, -3), B (5, 3) and C (3, -1)
Solution:

Question 23.
Three consecutive vertices of a parallelogram ABCD are A (1, 2), B (1, 0) and C (4, 0). Find the fourth vertex D.
Solution:

Question 24.
If the points A (-2, -1), B (1, 0), C (p, 3) and D (1, q) from a parallelogram ABCD, find the values of p and q.
Solution:

Question 25.
If two vertices of a parallelogram are (3, 2) (-1, 0) and its diagonals meet at (2, -5), find the other two vertices of the parallelogram.
Solution:

Question 26.
Prove that the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right-angled triangle. Find the coordinates of D so that ABCD is a square.
Solution:

Question 27.
Find the third vertex of a triangle if its two vertices are (-1, 4) and (5, 2) and the midpoint of one side is (0, 3).
Solution:

Question 28.
Find the coordinates of the vertices of the triangle the middle points of whose sides are \(\left( 0,\frac { 1 }{ 2 } \right) ,\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right) and\left( \frac { 1 }{ 2 } ,0 \right) \)
Solution:

Question 29.
Show by section formula that the points (3, -2), (5, 2) and (8, 8) are collinear.
Solution:

Question 30.
Find the value of p for which the points (-5, 1), (1, p) and (4, -2) are collinear.
Solution:

Question 31.
A (10, 5), B (6, -3) and C (2, 1) are the vertices of triangle ABC. L is the midpoint of AB, M is the mid-point of AC. Write down the co-ordinates of L and M. Show that LM = \(\\ \frac { 1 }{ 2 } \) BC.
Solution:

Question 32.
A (2, 5), B (-1, 2) and C (5, 8) are the vertices of a triangle ABC. P and.Q are points on AB and AC respectively such that AP : PB = AQ : QC = 1 : 2.
(i) Find the co-ordinates of P and Q.
(ii) Show that PQ = \(\\ \frac { 1 }{ 3 } \) BC.
Solution:

Question 33.
The mid-point of the line segment AB shown in the adjoining diagram is (4, -3). Write down die co-ordinates of A and B.

Solution:

Question 34.
Find the co-ordinates of the centroid of a triangle whose vertices are A (-1, 3), B(1, -1) and C (5, 1) (2006)
Solution:

Question 35.
Two vertices of a triangle are (3, -5) and (-7, 4). Find the third vertex given that the centroid is (2, -1).
Solution:

Question 36.
The vertices of a triangle are A (-5, 3), B (p, -1) and C (6, q). Find the values of p and q if the centroid of the triangle ABC is the point (1, -1).
Solution: