Question

A highway between points A and B has been closed for repairs. An alternative route between there two locations is to travel between A and C and then from C to B what is the value of Y and what is the total distance from A to C to B?

Answer 1

The value of Y is 24 km and distance from A to C to B is AC+Y=18+24=42 km

Explanation:

Given that A highway between points A and B has been closed for repairs. An alternative route between there two locations is to travel between A and C and then from C to B. we have to find the value of Y and the total distance from A to C to B. Let AB=Z

In ΔBCD and ΔABD

∠BCD=∠ABD (∵each 90°)

∠D=∠D (∵common)

By AA similarity, ΔBCD~ΔABD

∴ their corresponding sides are proportional

`(Y)/(Z)=(X)/(40)=(40)/(18+X)`

Comparing last two terms, we get

`(X)/(40)=(40)/(18+x)`

⇒
`X(18+X)=1600`

⇒
`18X+X^2=1600`

⇒
`X^2+18X-1600=0`

⇒
`X^2-32X+50X-1600=0`

⇒
`(X-32)(X+50)=0`

Hence, the roots are X=32, -50

X=-50 not possible as distance can never negative.

Hence, X=32 km

By applying Pythagoras theorem in ΔBCD we get

`BD^2=BC^2+CD^2`

`40^2=Y^2+32^2`

⇒
`Y^2=1600-1024=576`

⇒
`Y=\sqrt576=24km`

Hence, the value of Y is 24 km and the distance from A to C to B is AC+Y=18+24=42 km