8. Ashok wants to go by car from village A to village B.
He can (i) go straight from A to B (ii) first go to C due
east and then from C
to B due north. The
distance between A
and B exceeds the
distance between A
and C by 9 km. The
distance between A
and B is 9 km less
than twice the distance between B and C. Ashok
decides to go straight from A to B. Find the distance
between (i) A and B (ii) A and C (iii) C and B.
Answers
AB = 45 units.
AC = 36 units
And, CB = 27 units.
Step-by-step explanation:
See the attached diagram.
The distance between A and C = a and between C and B = b (say)
Now, the direct distance between A and B = \sqrt{a^{2} + b^{2}}.
From the given conditions we can write the equations
\sqrt{a^{2} + b^{2}} - a = 9 .......... (1)
And, 2b - 9 = \sqrt{a^{2} + b^{2}} ........ (2)
Now, from equations (1) and (2) we can write,
2b - 9 = 9 + a
⇒ 2b - a = 18 .......... (3)
Again, we can write from equations (1) and (2),
\sqrt{a^{2} + b^{2}} - a = 2b - \sqrt{a^{2} + b^{2}}
Squaring both sides we get,
a^{2} + b^{2} + a^{2} - 2a\sqrt{a^{2} + b^{2}} = 4b^{2} + a^{2} + b^{2} - 4y \sqrt{a^{2} + b^{2}}
⇒ a^{2} - 4b^{2} = 2\sqrt{a^{2} + b^{2}} (a - 2b)
⇒ a + 2b = 2\sqrt{a^{2} + b^{2}} {Since (a - 2b) ≠ 0, from equation (3)}
Again, squaring both sides we get,
a^{2} + 4b^{2} + 4ab = 4a^{2} + 4b^{2}
⇒ 4ab = 3a²
⇒ 4b = 3a {Since a ≠ 0}
⇒ a = \frac{4}{3}b ............... (4)
Now, solving equations (3) and (4) we get
2b - \frac{4}{3}b = 18
⇒ b = 27
And, a = \frac{4}{3}\times 27 = 36
Therefore, AB = \sqrt{36^{2} + 27^{2}} = 45 units.
AC = a = 36 units
And, CB = b = 27 units. (Answer)
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Given : Ashok wants to go by a car from village A to B.he can
(i) go straight from A to B
(ii) first go to C due east and then from C to B due north.
the distance between A and B exceeds the distance between A and C by 9 km.
the distance between A and B is 9km less than twice the distance between B and C.
To Find : distance between (i) A and B (ii) A and C (iii) C and B
Solution:
Distance between AC = x km
=> Distance between AB = x + 9
2 ( BC) - 9 = x + 9
=> 2 (BC) = x + 18
=> BC = x/2 + 9
AC² + BC² = AB²
=> x² + (x/2 + 9)² = (x + 9)²
=> x² + x²/4 + 9x + 81 = x² + 18x + 81
=> x²/4 = 9x
=> x/4 = 9
=> x = 36
Distance between AC = 36 km
Distance between AB = 36 + 9 = 45 km
Distance between BC = x/2 + 9 = (36/2) + 9 = 27 km
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