A, B, C, D and E are five friends standing in a park. The distance between A and C and that of B and C are same. E and C are 45m apart in west-east direction. A is towards west of C at a distance of 25m from E. C is in between B and D and in north direction wrt B.
Q.
What is the distance between A and B.
a)
26.2
b)
27.2
c)
28.2
d)
29.2
Correct answer is option 'C'. Can you explain this answer?

Question

A, B, C, D and E are five friends standing in a park. The distance between A and C and that of B and C are same. E and C are 45m apart in west-east direction. A is towards west of C at a distance of 25m from E. C is in between B and D and in north direction wrt B.

Q.
What is the distance between A and B.

a)
26.2
b)
27.2
c)
28.2
d)
29.2

Correct answer is option 'C'. Can you explain this answer?

Answer

Given information:
- A and C are at the same distance as B and C.
- E and C are 45m apart in the west-east direction.
- A is 25m west of C and B is north of C.
- C is between B and D in north direction with respect to B.

To find: The distance between A and B.

Solution:
1. Draw a rough diagram to represent the given information. Let's assume that C is at the origin (0,0) of the coordinate plane.
2. From the given information, we know that A is 25m to the left of C and E is 45m to the right of C. Therefore, the coordinates of A and E are (-25,0) and (45,0) respectively.
3. As A and B are equidistant from C, we can assume that B is at a point (x,y) such that it lies on a circle with C as the center and radius equal to the distance between A and C (let's call it R).
4. Using the distance formula, we can write:
- R = distance between A and C = sqrt((25+45)^2+0^2) = 70m (distance between A and E is 70m)
5. Now we can use the fact that C is between B and D in the north direction with respect to B to find the coordinates of B. Let's assume that the distance between B and C is d.
- Since C is at the origin, the y-coordinate of B is d.
- We also know that the distance between B and C is equal to R. Using the distance formula, we get:
(d-0)^2 + (x-0)^2 = R^2
d^2 + x^2 = R^2
x^2 = R^2 - d^2
x = sqrt(R^2 - d^2)
6. We still need to find the value of d to determine the coordinates of B. From the given information, we know that E and C are 45m apart in the west-east direction. Therefore, the x-coordinate of C is (45-0)/2 = 22.5m.
- Using the distance formula, we can write:
d^2 + (x-22.5)^2 = R^2
d^2 + (sqrt(R^2 - d^2) - 22.5)^2 = R^2
d^2 + (R^2 - 2d^2 + 506.25 - 45sqrt(R^2-d^2)) = R^2
d^2 - 2d^2 + 506.25 - 45sqrt(R^2-d^2) = 0
d^2 = (506.25 - 45sqrt(R^2-d^2))/(-1)
d^2 = 45sqrt(R^2-d^2) - 506.25
squaring both sides,
d^4 = 2025(R^2-d^2) - 506.25^2
d^4 + 2025d^2 - 2025R^2 + 506.25^2 = 0
On solving this quadratic equation, we get two values of d: d = 48.6m or d = 6.62m

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