Practice Set 5.2 Geometry 10th Std Maths Part 2 Answers Chapter 5 Co-ordinate Geometry
Question 1
Maharashtra Board Solution
Find the co-ordinates of point P if P divides the line segment joining the points A (-1, 7) and B (4, -3) in the ratio 2:3.
Solution & Step-by-Step Answer:
Let the co-ordinates of point P be (x, y) and A (x1, y1) B (x2, y2) be the given points. Here, x1 = -1, y1 = 7, x2 = 4, y2 = -3, m = 2, n = 3 ∴ By section formula, ∴ The co-ordinates of point P are (1,3).
Question 2
Maharashtra Board Solution
In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a : b. i. P (-3, 7), Q (1, -4), a : b = 2 : 1 ii. P (-2, -5), Q (4, 3), a : b = 3 : 4 iii. P (2, 6), Q (-4, 1), a : b = 1 : 2
Solution & Step-by-Step Answer:
Let the co-ordinates of point A be (x, y). i. Let P (x1, y1), Q (x2, y2) be the given points. Here, x1 = -3, y1 = 7, x2 = 1, y2 = -4, a = 2, b = 1 ∴ By section formula, ∴ The co-ordinates of point A are (,).
ii. Let P (x1,y1), Q (x2, y2) be the given points.
Here, x1= -2, y1= -5, x2= 4, y2= 3, a = 3, b = 4
By section formula,
∴ The co-ordinates of point A are (,)
iii. Let P (x1, y1), Q (x2, y2) be the given points.
Here,x1= 2,y1= 6, x2= -4, y2= 1, a = 1,b = 2
∴ By section formula,
∴ The co-ordinates of point A are (0,)
Question 3
Maharashtra Board Solution
Find the ratio in which point T (-1, 6) divides the line segment joining the points P (-3,10) and Q (6, -8).
Solution & Step-by-Step Answer:
Let P (x1, y1), Q (x2, y2) and T (x, y) be the given points. Here, x1 = -3, y1 = 10, x2 = 6, y2 = -8, x = -1, y = 6 ∴ By section formula, ∴ Point T divides seg PQ in the ratio 2 : 7.
Question 4
Maharashtra Board Solution
Point P is the centre of the circle and AB is a diameter. Find the co-ordinates of point B if co-ordinates of point A and P are (2, -3) and (-2,0) respectively.
Solution & Step-by-Step Answer:
Let A (x1, y1), B (x2, y2) and P (x, y) be the given points. Here, x1 = 2, y1 =-3, x = -2, y = 0 Point P is the midpoint of seg AB. ∴ By midpoint formula, ∴ The co-ordinates of point B are (-6,3).
Question 5
Maharashtra Board Solution
Find the ratio in which point P (k, 7) divides the segment joining A (8, 9) and B (1,2). Also find k.
Solution & Step-by-Step Answer:
Let A (x1, y1), B (x2, y2) and P (x, y) be the given points. Here, x1 = 8, y1 = 9, x2 = 1, y2 = 2, x = k, y = 7 ∴ By section formula, ∴ Point P divides seg AB in the ratio 2 : 5, and the value of k is 6.
Question 6
Maharashtra Board Solution
Find the co-ordinates of midpoint of the segment joining the points (22, 20) and (0,16).
Solution & Step-by-Step Answer:
Let A (x1, y1) = A (22, 20), B (x2,y2) = B (0, 16) Let the co-ordinates of the midpoint be P (x,y). ∴ By midpoint formula, The co-ordinates of the midpoint of the segment joining (22, 20) and (0, 16) are (11,18).
Question 7
Maharashtra Board Solution
Find the centroids of the triangles whose vertices are given below. i. (-7, 6), (2,-2), (8, 5) ii. (3, -5), (4, 3), (11,-4) iii. (4, 7), (8, 4), (7, 11)
Solution & Step-by-Step Answer:
i. Let A (x1, y1) = A (-7, 6), B (x2, y2) = B (2, -2), C (x3, y3) = C(8, 5) ∴ By centroid formula, ∴ The co-ordinates of the centroid are (1,3).
ii. Let A (x1y1) = A (3, -5),
B (x2, y2) = B (4, 3),
C(x3, y3) = C(11,-4)
∴ By centroid formula,
∴ The co-ordinates of the centroid are (6, -2).
iii. Let A (x1, y1) = A (4, 7),
B (x2, y2) = B (8,4),
C (x3, y3) = C(7,11)
∴ By centroid formula,
∴ The co-ordinates of the centroid are (,)
Question 8
Maharashtra Board Solution
In ∆ABC, G (-4, -7) is the centroid. If A (-14, -19) and B (3, 5), then find the co-ordinates of C.
Solution & Step-by-Step Answer:
G (x, y) = G (-4, -7), A (x1, y1) = A (-14, -19), B(x2, y2) = B(3,5) Let the co-ordinates of point C be (x3, y3). G is the centroid. By centroid formula, ∴ The co-ordinates of point C are (-1, – 7).
Question 9
Maharashtra Board Solution
A (h, -6), B (2, 3) and C (-6, k) are the co-ordinates of vertices of a triangle whose centroid is G (1,5). Find h and k.
Solution & Step-by-Step Answer:
A(x1,y1) = A(h, -6), B (x2, y2) = B(2, 3), C (x3, y3) = C (-6, k) ∴ centroid G (x, y) = G (1, 5) G is the centroid. By centroid formula, ∴ 3 = h – 4 ∴ h = 3 + 4 ∴ h = 7 ∴ 15 = -3 + k ∴ k = 15 + 3 ∴ k = 18 ∴ h = 7 and k = 18
Question 10
Maharashtra Board Solution
Find the co-ordinates of the points of trisection of the line segment AB with A (2,7) and B (-4, -8).
Solution & Step-by-Step Answer:
A (2, 7), B H,-8) Suppose the points P and Q trisect seg AB. ∴ AP = PQ = QB ∴ Point P divides seg AB in the ratio 1:2. ∴ By section formula, Co-ordinates of P are (0, 2). Point Q is the midpoint of PB. By midpoint formula, Co-ordinates of Q are (-2, -3). ∴ The co-ordinates of the points of trisection seg AB are (0,2) and (-2, -3).
Question 11
Maharashtra Board Solution
If A (-14, -10), B (6, -2) are given, find the co-ordinates of the points which divide segment AB into four equal parts.
Solution & Step-by-Step Answer:
Let the points C, D and E divide seg AB in four equal parts. Point D is the midpoint of seg AB. ∴ By midpoint formula, ∴ Co-ordinates of D are (-4, -6). Point C is the midpoint of seg AD. ∴ By midpoint formula, ∴ Co-ordinates of C are (-9, -8). Point E is the midpoint of seg DB. ∴ By midpoint formula, ∴ Co-ordinates of E are (1,-4). ∴ The co-ordinates of the points dividing seg AB in four equal parts are C(-9, -8), D(-4, -6) and E(1, – 4).
Question 12
Maharashtra Board Solution
If A (20, 10), B (0, 20) are given, find the co-ordinates of the points which divide segment AB into five congruent parts.
Solution & Step-by-Step Answer:
Suppose the points C, D, E and F divide seg AB in five congruent parts. ∴ AC = CD = DE = EF = FB ∴ co-ordinates of C are (16, 12). E is the midpoint of seg CB. By midpoint formula, ∴ co-ordinates of E are (8, 16). D is the midpoint of seg CE. ∴ co-ordinates of F are (4, 18). ∴ The co-ordinates of the points dividing seg AB in five congruent parts are C (16, 12), D (12, 14), E (8, 16) and F (4, 18).
Maharashtra Board Class 10 Maths Chapter 5 Co-ordinate Geometry Intext Questions and Activities
Question 1
Maharashtra Board Solution
A (15, 5), B (9, 20) and A-P-B. Find the ratio in which point P (11, 15) divides segment AB. Find the ratio using x and y co-ordinates. Write the conclusion. (Textbook pg. no. 113)
Solution & Step-by-Step Answer:
Suppose point P (11,15) divides segment AB in the ratio m : n. By section formula, ∴ Point P divides seg AB in the ratio 2 : 1. The ratio obtained by using x and y co-ordinates is the same.
Question 2. External division: (Textbook pg. no. 115) Suppose point R divides seg PQ externally in the ratio 3:1. Let the common multiple be k. Let PR = 3k and QR = k Now, PR = PQ + QR … [P – Q – R] ∴ 3k = PQ + k ∴ = = ∴ Point Q divides seg PR in the ratio 2 : 1 internally. Thus, we can find the co-ordinates of point R, when co-ordinates of points P and Q are given. |