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Problem 7 Easy

Frances stands 15 meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located x meters east of the locked gate. An unlocked gate lies 9 meters east of the box, and another unlocked gate lies 8 meters west of the locked gate. Frances can reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel would be the same via either unlocked gate. What is value of x?

  • A.

    3 \frac { 2 } { 7 }

  • B.

    3 \frac { 3 } { 7 }

  • C.

    3 \frac { 4 } { 7 }

  • D.

    3 \frac { 5 } { 7 }

  • E.

    3 \frac { 6 } { 7 }

Answer:C
Solution

Let G_2F = y. Here, FL \perp G_1G_2, BL = x, BG_2 = 9, LG_1 = 8, LF = 15.

Then G_1F = \sqrt{LF^2 + LG_1^2} = \sqrt{8^2 + 15^2} = 17

BG_1 + FG_1 = 8 + x + 17 = 25 + x

Also, 9 + y = 25 + x, which gives y = 16 + x.

Since y^2 = (x+9)^2 + 15^2, we have (16+x)^2 = (x+9)^2 + 15^2, which gives x = \frac{25}{7}.

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