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Two sprinklers, Q and R, are located 5x/6 feet apart in a rectangular lawn bordered on one side by a straight fence of negligible thickness. Sprinkler Q sprays water in circle with a radius of 5x/3 feet and is x feet from the fence. Sprinkler R sprays water in a circle with a radius of 5x/2 feet and is 3x/2 feet from the fence. What is the length of the part of the fence that is sprayed by the sprinklers?
Below is the link of the original question posted by franteraoka
https://gmatclub.com/forum/two-sprinklers-q-and-r-are-located-5x-6-feet-apart-in-a-rectangular-270734.html
2x
3x
4x
5x
6x
OE :
Plug in. Let x=6
Draw a horizontal line representing the fence. Place point Q 6 feet below the fence, and draw a circle around point Q with a diameter of 20 feet. Then, place point R 5 feet from point Q and 9 feet below the fence, and draw a circle around point R with a diameter of 30 feet, so that the circle with center Q lies within the circle with center R. The question asks for the length of fence that is sprayed by the sprinklers, which is the distance between two points. To find the distance between two points, use right triangles.
Draw a line from point R to the fence such that the line and the fence are perpendicular. Draw another line from point R to the point where the circumference of the circle with center R intersects the fence furthest from point Q. This creates a right triangle with one leg of length 9 and a hypotenuse of length 15. A right triangle with a hypotenuse of 15 and a leg of 9 is a Pythagorean triple, so the side of the triangle adjacent to the fence is of length 12. Draw an analogous triangle for point Q that has a leg of length 6 and a hypotenuse of 10, which intersects the fence at the point on the circumference of the circle with center Q furthest from point R. This triangle is also a Pythagorean triple, so the side of the triangle adjacent to the fence is of length 8. Now, determine the length of fence that separates the two points on the fence.
Draw a line connecting point R to the fence at a 90° angle. Draw a horizontal line from point Q to the line connecting point R to the fence. Draw a line connecting points Q and R. This creates a triangle with a hypotenuse of 5 and a leg of 9 – 6 = 3. This, too, is a Pythagorean triple, so the third side of the triangle is of length 4. Since the third side of this triangle is equal in length to the portion of fence that separates points Q and R, the length of fence that is sprayed by the sprinklers is 8 + 4 + 12 = 24. Circle the target answer, 24.
Now, plug x = 6 into each of the answer choices, looking for the choice that equals 24. Since 4(6) = 24, the correct answer is choice C.
Below is the link of the original question posted by franteraoka
https://gmatclub.com/forum/two-sprinklers-q-and-r-are-located-5x-6-feet-apart-in-a-rectangular-270734.html
2x
3x
4x
5x
6x
OE :
Plug in. Let x=6
Draw a horizontal line representing the fence. Place point Q 6 feet below the fence, and draw a circle around point Q with a diameter of 20 feet. Then, place point R 5 feet from point Q and 9 feet below the fence, and draw a circle around point R with a diameter of 30 feet, so that the circle with center Q lies within the circle with center R. The question asks for the length of fence that is sprayed by the sprinklers, which is the distance between two points. To find the distance between two points, use right triangles.
Draw a line from point R to the fence such that the line and the fence are perpendicular. Draw another line from point R to the point where the circumference of the circle with center R intersects the fence furthest from point Q. This creates a right triangle with one leg of length 9 and a hypotenuse of length 15. A right triangle with a hypotenuse of 15 and a leg of 9 is a Pythagorean triple, so the side of the triangle adjacent to the fence is of length 12. Draw an analogous triangle for point Q that has a leg of length 6 and a hypotenuse of 10, which intersects the fence at the point on the circumference of the circle with center Q furthest from point R. This triangle is also a Pythagorean triple, so the side of the triangle adjacent to the fence is of length 8. Now, determine the length of fence that separates the two points on the fence.
Draw a line connecting point R to the fence at a 90° angle. Draw a horizontal line from point Q to the line connecting point R to the fence. Draw a line connecting points Q and R. This creates a triangle with a hypotenuse of 5 and a leg of 9 – 6 = 3. This, too, is a Pythagorean triple, so the third side of the triangle is of length 4. Since the third side of this triangle is equal in length to the portion of fence that separates points Q and R, the length of fence that is sprayed by the sprinklers is 8 + 4 + 12 = 24. Circle the target answer, 24.
Now, plug x = 6 into each of the answer choices, looking for the choice that equals 24. Since 4(6) = 24, the correct answer is choice C.
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