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# Starting from point O on a flat school playground, a child walks 10 ya

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Math Expert
Joined: 02 Sep 2009
Posts: 46068
Starting from point O on a flat school playground, a child walks 10 ya [#permalink]

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05 Oct 2017, 08:04
00:00

Difficulty:

25% (medium)

Question Stats:

96% (00:46) correct 4% (00:52) wrong based on 28 sessions

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Starting from point O on a flat school playground, a child walks 10 yards due north, then 6 yards due east, and then 2 yards due south, arriving at point P. How far apart, in yards, are points O and P?

(A) 18
(B) 16
(C) 14
(D) 12
(E) 10

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Posts: 1745
Starting from point O on a flat school playground, a child walks 10 ya [#permalink]

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05 Oct 2017, 10:14
Bunuel wrote:
Starting from point O on a flat school playground, a child walks 10 yards due north, then 6 yards due east, and then 2 yards due south, arriving at point P. How far apart, in yards, are points O and P?

(A) 18
(B) 16
(C) 14
(D) 12
(E) 10

Attachment:

iiiiii.png [ 5.51 KiB | Viewed 438 times ]

The distance in yards between points O and P is 10

The child creates a trapezoid with her walking pattern. The trapezoid can be divided into a rectangle and a right triangle

1) Find shape created by walking pattern. It will contain right angles (with compass points, "due [any direction]" = 90 degrees)

She walks 10 yards due north from point O to point A: OA = 10

Then 6 yards due east from A to B, which means a right angle at A. AB = 6

Then 2 yards due south from B to P. Another right angle. BP = 2

Connect P to O. She has created a trapezoid with her travel

2) Divide the shape into other shapes where distance between start and finish is easy to calculate

Draw a line from P, PX on diagram, that is perpendicular to OA
The new line divides trapezoid into two new shapes. Upper region is a rectangle. Lower region is right triangle

Rectangle has length = 6, width = 2

The rectangle's width, BP = AX = 2, divides original OA = 10 into two lengths AX and OX:
(OA - AX) = OX
(10 - 2) = 8 = OX

Looking at leg lengths PX and OX: the bottom portion is a 3-4-5 right triangle

3) Find distance between begin and end point

Find length of hypotenuse OP
Rule: If a right triangle has two legs in ratio 3x:4x, it is a 3x-4x-5x triangle and the hypotenuse = 5x

Leg PX = 6
Leg OX = (10 - 2) = 8

(3x: 4x: 5x) = (6: 8: 10)*

Distance between points O and P, in yards: 10

*Or use Pythagorean theorem:
$$6^2 + 8^2 = OP^2$$
$$36 + 64 = OP^2$$
$$100 = OP^2$$
$$OP = 10$$
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Starting from point O on a flat school playground, a child walks 10 ya   [#permalink] 05 Oct 2017, 10:14
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