Question

Read the following passage:

Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C.

Based on the above information, answer the following questions.

1. How far is Alia's house from Shagun's house?
2. How far is the library from Shagun's house?
3. Show that for Shagun, school is farther compared to Alia's house and library.
   OR
   Show that Alia’s house, shagun’s house and library for an isosceles right triangle.

Answer 1

Coordinates of A (2, 3) – Alia's house

Coordinates of B (2, 1) – Shagun's house

Coordfuates of C (4, 1) – Library

i. AB = ${\displaystyle \sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}}$

= ${\displaystyle \sqrt{{(2-2)}^2+{(1-3)}^2}}$

= ${\displaystyle \sqrt{{\left(0\right)}^2+{(-2)}^2}}$

AB = ${\displaystyle \sqrt{0+4}}$

= ${\displaystyle \sqrt{4}}$ unit

= 2 units

Alia's house from shagun's house is 2 units

ii. C(4, 1), B(2, 1)

CB = ${\displaystyle \sqrt{{(2-4)}^2+{(1-1)}^2}}$

= ${\displaystyle \sqrt{{(-2)}^2+0^2}}$

= ${\displaystyle \sqrt{4+0}}$

= ${\displaystyle \sqrt{4}}$

= 2 unit

iii. O(0, 0), B(2, 1)

OB = ${\displaystyle \sqrt{{(2-0)}^2+{(1-0)}^2}}$

= ${\displaystyle \sqrt{2^2+1^2}}$

= ${\displaystyle \sqrt{4+1}}$

= ${\displaystyle \sqrt{5}}$ units

Distance between Alia's house and Shagun's house, AB = 2 units

Distance between Library and Shagun's house, CB = 2 units

OB is greater than AB and CB,

For shagun, school [O] is farther than Alia's house [A] and Library [C]

OR

C(4, 1), A(2, 3)

CA = ${\displaystyle \sqrt{{(2-4)}^2+{(3-1)}^2}}$

= ${\displaystyle \sqrt{{(-2)}^2+2^2}}$

= ${\displaystyle \sqrt{4+4}}$

= ${\displaystyle \sqrt{8}}$

= ${\displaystyle 2\sqrt{2}}$ units

AC² = 8

Distance between Alia’s house and Shagun’s house, AB = 2 units

Distance between Library and Shagun’s house, CB = 2 units

AB² + BC²

= 2² + 2²

= 4 + 4

= 8

= AC² 

Therefore A, B and C form an isosceles right triangle.