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Geometry Points of Concurrency HW Worksheet Name:________________________ For problems 1-4 identify the point of concurrency shown and what constructions form it: 1. Point: ______________ Formed By: ____________ ______________________ 2. Point: _______________ Formed By: _____________ _______________________ 3. Point: _______________ Formed By: _____________ _______________________ 4. Point: _______________ Formed By:_____________ _______________________ 5. Point G is the centroid of ∆ABC. ̅̅̅̅ 𝐴𝐷 = 8, ̅̅̅̅ 𝐴𝐺 = 10, 𝑎𝑛𝑑 ̅̅̅̅ 𝐶𝐷 = 18. Find the length of the given segments: ̅̅̅̅ =_____________ 𝐵𝐷 ̅̅̅̅ 𝐴𝐸 =_____________ ̅̅̅̅ 𝐴𝐵 =_____________ 𝐶𝐺 =_____________ ̅̅̅̅ 𝐸𝐺 =_____________ ̅̅̅̅ =_____________ 𝐷𝐺 ̅̅̅̅ = 10, 𝐶𝐹 ̅̅̅̅ = 12. 6. D is the centroid of ∆ABC, ̅̅̅̅ 𝐴𝐸 = 12, 𝐴𝐷 Find the length of each segment: ̅̅̅̅ 𝐷𝐺 =____________ ̅̅̅̅ 𝐴𝐺 =_____________ ̅̅̅̅ =___________ 𝐷𝐹 ̅̅̅̅ =____________ 𝐸𝐶 ̅̅̅̅ 𝐴𝐶 =_____________ 7. Which Point of Concurrency is the center of an inscribed circle as shown below? ____________________________________ 8. Which Point of Concurrency is the center of a circumscribed circle as shown below? _______________________________ For Problems 9-21, write the letter that corresponds to the point of concurrency in the space provided. A. Circumcenter B. Incenter C. Centroid D. Orthocenter 9. Which points of concurrency are always inside the triangle? ____________________, __________________ 10. Which point of concurrency is always on the vertex of a right triangle? ______________________________ 11. Which point of concurrency is always on the midpoint of the hypotenuse in a right triangle? _____________ 12. Which points of concurrency are always outside of an obtuse triangle? _______________, ______________ 13. Which point of concurrency is the center of gravity in a triangle? ___________________________ 14. Which point of concurrency is equidistant from every vertex? _____________________________ 15. The three altitudes of a triangle intersect at the ____________________. 16. The three medians of a triangle intersect at the ____________________. 17. The three perpendicular bisectors of a triangle intersect at the ____________________. 18. The three angle bisectors of a triangle intersect at the ____________________. 19. It is equidistant from the three vertices of the triangle. _______________ 20. It is equidistant from the three sides of the triangle. _________________ 21. It divides each median into two sections at a 2:1 ratio. _______________ For Problems 22-24 State a point of concurrency that would help solve each of the problems below. Then state how you would find that point of concurrency. 22. This rectangle represents the Jones’ lot. The non-shaded triangular region represents their backyard. The Jones’ want to build the largest possible circular pool in their back yard, how would you determine the location of the pool’s center? Merry-go-round 23. The Smith Construction Company has been hired to install a new water fountain at Winstonian Park. They would like to find the best location for the fountain so that the walking distance from each of the three main pieces of playground equipment is the same. How would they determine this point? 24. You are a sculptor and have just completed a large metal mobile. You want to hang this mobile, made of a flat triangular metal plate, in the State Capitol. This triangular piece will hang so that it will be suspended with the triangular surface parallel to the ground. How would you locate the point where the mobile will balance? Name the point of concurrency shown for the bold triangle. 25. ____________________ 26. __________________ Swing 27. ______________________ See-Saw