Question

In a cinema hall, peoples are seated at a distance of 1 m from each other, to maintain the social distance due to CORONA virus pandemic. Let three peoples sit at points P, Q and R whose coordinates are (6,-2), (9, 4) and (10, 6) respectively. Based on the above information, answer the following questions. (i) The distance between P and Ris (a) √5 units (b)4√5 units (c)3√5 units (d) None of these (ii) If a TC at the point I, lying on the straight line joining Q and R such that it divides the distance between them in the ratio of 1: 2. Then coordinates of I are (c)(6, 1) (d)(9.1) (iii) The mid-point of the line segment joining P and Ris (a)(1,6) (b)(6.1) (d) None of these (iv) The ratio in which Q divides the line segment joining P and Ris (a)2:1 (c)1:2 (b)3:1 (d) None of these (v) The points P, Q and R lies on (a) a straight line (b) an equilateral triangle (c) a scalene triangle (d) an isosceles triangle​

Answer 1

Step-by-step explanation:

(a) √5 units

is the corect answer ..

Answer 2

Given:

Coordinates of P, Q, R are (6,-2), (9, 4), and (10, 6), respectively

To find:

i. Distance between P and R

ii. Coordinates of I when it divides QR in 1:2

iii. Mid-point of PR

iv. Ratio in which Q divides PR

v. Figure formed by P, Q, and R.

Solution:

We can find the solution by following the given process-

i. We know that the distance between any two points can be calculated as the root of the sum of x and y coordinates' difference.

Coordinates of P (x1, y1)= (6, -2)

Coordinates of R (x2, y2)= (10, 6)

The distance between P and R=

$\sqrt{ {(x2 - x1)}^{2} + {(y2 - y1)}^{2} }$

$PR= \sqrt{ {(10 - 6)}^{2} + {(6 - ( - 2))}^{2} }$

$PR= \sqrt{ {4}^{2} + {8}^{2} }$

$PR= \sqrt{16 + 64}$

$PR= \sqrt{80}$

$PR=4 \sqrt{5} \: units$

ii. The point I divide QR in the ratio of m:n, where m:n=1:2

The coordinates of the point I =(mx2+nx1)/m+n, (my2+ny1)/m+n

Coordinates of Q, (x1, y1)= (9, 4)

Coordinates of R, (x2, y2)= (10, 6)

Coordinates of I, (x3, y3), are as follows-

x3= (mx2+nx1)/m+n

= (1×10+2×9)/3

=28/3

y3=(my2+ny1)/m+n

=(1×6+2×4)/3

=14/3

Coordinates of I= (28/3, 14/3)

iii. Mid-point of a line divides it in the ratio of 1:1.

The coordinates of mid-point of PR= (x1+x2)/2, (y1+y2)/2

x1, y1, x2, y2 are the coordinates of P and R.

Mid points= (6+10)/2, (6-2)/2

Mid-points of PR=(8, 2)

iv. Let the ratio in which Q divides PR be m:n.

So, the coordinates of Q divide PR in m:n.

Coordinates of P (x1, y1)= (6, -2)

Coordinates of R (x2, y2)= (10, 6)

Coordinates of Q, (x3, y3)= (9, 4)

x3= (mx2+nx1)/m+n

9=(m×10+n×6)/m+n

9(m+n)= 10m+6n

9m+ 9n= 10m+6n

3n=m

m/n=3/1

m:n=3:1

v. We can calculate the length of all sides of the triangle formed to determine the figure formed by P, Q, and R.

PR=4√5 units

Similarly,

$PQ= \sqrt{ {(9 - 6)}^{2} + {(4 + 2)}^{2} }$

$PQ = \sqrt{9 + 36}$

$PQ = \sqrt{45}$

PQ=3√5 units

and

$QR = \sqrt{ {(10 - 9)}^{2} + {(6 - 4)}^{2} }$

$QR = \sqrt{1 + 4}$

QR=√5 units

Since all three sides are unequal, it is a scalene triangle.

Therefore, PR is 4√5 units, Coordinates of I are (28/3, 14/3), Mid-points of PR are (8, 2), the ratio in which Q divides PR is 3:1, and P, Q, R form a scalene triangle.