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Ex 7.2, 2 Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3). Let the given points be A (4, −1) & B (−2, 3) P & Q are two points on AB such that AP = PQ = QB Let AP = PQ = QB = m Point P divides AP & PB in the ratio AP = m PB = PQ + QB = k + k = 2k Hence, Ratio between AP & PB = AP/PB = 𝑘/2𝑘 = 1/2 Thus P divides AB in the ratio 1:2 Finding P Let P(x, y) m1 = 1, m2 = 2 And for AB x1 = 4, x2 = −2 y1 = −1, y2 = −3 x = (𝑚_1 𝑥_2 + 𝑚_2 𝑥_1)/(𝑚_1 + 𝑚_2 ) = (1 ×−2 + 2 × 4)/(1 + 2) = (−2 + 8)/3 = 6/3 = 2 y = (𝑚_1 𝑦_(2 )+ 𝑚_2 𝑦_1)/(𝑚_1 + 𝑚_2 ) = (1 × −3 + 2 × −1)/(1 + 2) = (−3 − 2)/3 = (−5)/3 Hence, point P is P(x, y) = P ("2, " (−5)/3) Similarly, Point Q divides AB in the ratio AQ & QB 𝐴𝑄/𝑄𝐵 = (𝐴𝑃 + 𝑃𝑄)/𝑄𝐵 = (𝑘 + 𝑘 )/𝑘 = 2𝑘/𝑘 = 2/1 = 2 : 1 Finding Q Let Q be Q(x , y) m1 = 2, m2 = 1 x1 = 4, x2 = −2 y1 = −1, y2 = −3 x = (𝑚_1 𝑥_2 + 𝑚_2 𝑥_1)/(𝑚_1 + 𝑚_2 ) = (2 ×−2 + 1 × 4)/(2 + 1) = (−4 + 4)/3 = 0/3 = 0 y = (𝑚_1 𝑦_2 + 𝑚_2 𝑦_1)/(𝑚_1 + 𝑚_2 ) = (2 × −3 + 1 × −1)/(1 + 2) = (−6 − 1)/3 = (−7)/3 Hence, point Q is Q(x, y) = Q ("0, " (−𝟕)/𝟑)
Solution:
The coordinates of the point P(x, y) which divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂) internally in the ratio m₁: m₂ is given by the section formula.
Let the points be A(4, - 1) and B(- 2, - 3).
Let P (x₁, y₁) and Q (x₂, y₂) be the points of trisection of the line segment joining the given points.
Then, AP = PC = CB
By Section formula ,
P(x, y) = [(mx₂ + nx₁) / (m + n), (my₂ + ny₁) / (m + n)] ...... (1)
Considering A(4, - 1) and B(- 2, - 3), by observation point P(x₁, y₁) divides AB internally in the ratio 1 : 2.
Hence m : n = 1 : 2
By substituting the values in the Equation (1)
x₁ = [1 × (- 2) + 2 × 4] / (1 + 2)
x₁ = (- 2 + 8) / 3
= 2
y₁ = [1 × (- 3) + 2 × (- 1)] / (1 + 2)
y₁ = (- 3 - 2) / (1 + 2)
= - 5/3
Hence, P(x₁ , y₁) = (2, - 5/3)
Now considering A(4, - 1) and B(- 2, - 3), by observation point C(x₂, y₂) divides AB internally in the ratio 2 : 1.
Hence m : n = 2 : 1
By substituting the values in the Equation (1)
x₂ = [2 × (- 2) + 1 × 4] / (2 + 1)
= (- 4 + 4) / 3
= 0
y₂ = [2 × (- 3) + 1 × (- 1)] / (2 + 1)
= (- 6 - 1) / 3
= - 7/3
Therefore, C(x₂ , y₂) = (0, - 7/3)
Hence, the points of trisection are P(x₁ , y₁) = (2, - 5/3) and C (x₂ , y₂) = (0, - 7/3)
☛ Check: NCERT Solutions Class 10 Maths Chapter 7
Video Solution:
Find the coordinates of the points of trisection of the line segment joining (4, - 1) and (- 2, - 3).
NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 2
Summary:
The coordinates of the points of trisection of the line segment joining (4, - 1) and (- 2, - 3) are (2, - 5/3) and (0, - 7/3).
☛ Related Questions:
- 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 with AD, as shown in Fig. 7.12. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5 th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segments joining the two flags, where should she post her flag?
- Find the ratio in which the line segment joining the points (- 3, 10) and (6, - 8) is divided by (- 1, 6).
- 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.
- If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Text Solution
Solution : Let P `(a, b) and Q(c, d)` trisect the line joining the points `A(3, -2) and B(-3, -4)`. <br> Now, point P (a, b) divides the line AB in the ratio `1: 2`. <br> <img src="//d10lpgp6xz60nq.cloudfront.net/physics_images/NTN_MATH_X_C07_S01_035_S01.png" width="80%"> <br> `therefore" "a=(1(-3)+2(3))/(1+2)=(3)/(3)=1` <br> `" "b=(1(-4)+2(-2))/(1+2)=-(8)/(3)` <br> Therefore, co-ordinates of point P = `(1, -(8)/(3))` <br> `Q(c, d)` divides the line AB in the ratio 2 :1. <br> `therefore" "c=(2(-3)+1(3))/(2+1)=-1` <br> `" "d=(2(-4)+1(-2))/(2+1)=(-10)/(3)` <br> Therfore, co-ordinates of `Q=(-1, (-10)/(3))` <br> `therefore` Co-ordinates of points of trisection of AB = `=(1, (-8)/(3))and (-1, (-10)/(3))`
Find the points of trisection of the line segment joining the points A 4,3 and B 2, 1.A. 2, 5/3 and 0, 1/3B. 2, 5/3 and 2, 1/3C. 2, 5/3 and 0, 1/3D. 2, 5/3 and 0, 4/3
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