Task on Points of Concurrency with Hospitals
Task on Points of Concurrency with Hospitals Name______
Universal Health Services (UHS) is a company that owns and operates over 100 hospitals in the United States.
Suppose that UHS wishes to construct one hospital to provide service to the cities of Metropolis, Gotham City, and Quahog because of the high number of injuries in these three places. In addition, suppose the three towns are already connected by three different highways.
The three cities and their interconnected highways form a right triangle as indicated on the answer sheet. Using the directions below, label the remaining vertices and sides of the right triangle on the answer sheet. Quahog has already been labeled with the letter Q, and Highway 57 has already been labeled
as H-57.
1. Label Gotham City with the letter G - it is located at the vertex of the right angle.
2. Label Metropolis with the letter M - it is located at the last remaining vertex.
3. Label Highway 23 as H-23 - it is the hypotenuse of the right triangle.
4. Label Highway 88 as H-88 - it is last remaining unnamed side.
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5. Fill in the blanks on the answer sheet: The circumcenter is the point of concurrency
for the ______of a triangle.
6. Fill in the blanks on the answer sheet: An important property of the circumcenter of a triangle is that it is equidistant from all three ______of a triangle. The word equidistant means "the same distance away".
7. Suppose United Health Services elects to build the hospital at the circumcenter.
A)Place a point at the circumcenter of the right triangle on the answer sheet. Label this point with the letter C.
B)Using the property of the circumcenter from Question 6, write a statement using complete sentence(s) about why it would be beneficial for UHS to construct the hospital at the circumcenter.
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8. On the right triangle on the answer sheet, suppose the vertex of the angle with the largest
measure is the location of the city with the largest population, and the vertex of the angle
with the smallest measure is the location of the city with the smallest population. Which city has the largest population?
9. If the distance from Quahog to Gotham City is 40 miles, and the distance from Metropolis to Gotham City is 75 miles, what is the distance from Quahog to Metropolis? Hint - you must
use a "theorem" to get the answer. Show all work required in the space provided.
10. Using the information in Questions 8-9, which city must have the smallest population? Describe how you obtained your answer. Use complete sentences. Your answer cannot be anything
like, "That angle looked like it was the smallest."
11. Suppose UHS decides to build the hospital on the midpoint of Highway 57.
A)In miles, how far would the hospital be from Quahog?
B)For easy access to this hospital, a road must be built connecting Metropolis to the hospital. Which of the following vocabulary words would most accurately describe this road:
altitude, perpendicular bisector, angle bisector, median, or hypotenuse?
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12. Fill in the blanks on the answer sheet: The orthocenter is the point of concurrency for the ______of a triangle.
13. Suppose United Health Services elects to build the hospital at the orthocenter.
A)Place a STAR at the location of the orthocenter for the right triangle on the answer sheet.
Hint - the fact that it is a right triangle means the orthocenter is at a special location.
B)Using the information from Question 8!!!, write a statement using complete sentence(s) about why it would be beneficial for UHS to construct the hospital at the orthocenter.
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Important Discovery - After doing some research of the cities and the surrounding areas, United Health Services determines that more people live along each of the highways than in the cities themselves.
14. Fill in the blanks on the answer sheet: The incenter is the point of concurrency for the ______of a triangle.
15. Fill in the blanks on the answer sheet: An important property of the incenter of a triangle is that it is equidistant from all three ______of a triangle.
16. Suppose United Health Services elects to build the hospital at the incenter.
A)Place a point at the approximate location of the incenter of the right triangle on the answer sheet. Label this point with the letter I.
B)Using the information from the Important Discovery and from Question 15, write a statement using complete sentence(s) about why it would be beneficial for UHS to construct the hospital at the incenter.
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The name for the city of Metropolis came from the Superman comic book series. In this series, a world called Bizarro World exists that takes everything and flips it upside down. For example, in Bizarro World, the citizens live on the planet "HTRAE" instead of EARTH. Instead of drinking milk, they drink "klim".
In Bizarro World, suppose the three cities of Siloportem, Ytic Mahtog, and Gohauq, along with their three interconnected highways, also form a right triangle. Let Ytic Mahtog be located at the vertex of the right angle of the right triangle.
17. Sketch a picture of this right triangle, labeling the vertices with the letters S, Y , and G for the first
letters of the cities they represent. Also, make a marking to show the location of the right angle.
18. Suppose that YG = 75 miles, and SG = 85 miles. Would this right triangle formed in Bizarro World have to be congruent to the one referenced at the very beginning of the entire task? Answer Yes or No. If "Yes", by what theorem taught in class would it be best to prove that the two triangles are congruent?
Answer Sheet to Task on Points of Concurrency Name______
and Hospitals
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9. Show all work in the space below and circle the answer.
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17. Sketch the picture in the space below.
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