Practice 1.9.4: Proving Centers of Triangles
Use what you know about centers of triangles to complete each problem.
1. Triangle ABC has vertices A (0, 0), B (0, –6), and C (–2, 0). Justify that (–1, –
3) is the circumcenter of Triangle ABC .
2. Triangle ABC has vertices A (–2, 6), B (–2, 1), and C (4, 1). Justify that (–
2, 1) is the orthocenter of Triangle ABC .
3. Triangle ABC has vertices A (7, 3), B (8, –4), and C (0, –2). Justify that (5,
–1) is the centroid of Triangle ABC .
4. Verify that the centroid, (5, –1), of Triangle ABC with vertices A (7, 3), B (8, –
4), and
2
C (0, –2) is the distance from each vertex to the midpoint of the opposite side.
3
5. Triangle ABC has vertices A (2, 8), B (0, 9), and C (5, –1). Will the incenter be
inside, outside, or on a side of Triangle ABC ? Explain your answer.
6. Triangle ABC has vertices A (–3, 7), B (2, 7), and C (2, –4). Will the
orthocenter be inside, outside, or on a side of Triangle ABC ? Explain your
answer.
continued
PRACTICE
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 9: Proving Theorems About Triangles
7. The Circumcenter Theorem states the circumcenter of a triangle is equidistant
from the vertices of the triangle. Prove this theorem using the information below.
Given: Triangle ABC has perpendicular bisectors p, q, and r of AB , BC ,
and AC . Prove: AX = BX = CX
B
q
r
p
A
C
X
8. A trauma center is to be built to assist three towns. The relative location of the
towns and the concurrent roads connecting the towns are shown below. If the
trauma center cannot be built outside the area of the triangle, which point(s) of
concurrency cannot be used to determine the location of the trauma center?
continued
PRACTICE
UNIT 1 • SIMILARITY, CONGRUENCE, AND PROOFS
Lesson 9: Proving Theorems About Triangles
9. A new first aid station at a local park is to be placed in a location that is equidistant
from the information booth, the picnic area, and the water fountain. Which center
of the triangle created between each location should be determined? Explain your
answer.
10. A dog’s leash is staked in a triangular backyard. Which center of the yard should be
found to ensure the dog has the maximum amount of space without going into the
neighbor’s yard?