Chapter 5 Test 1 Review
A. Construction of Circumcenter (circumcircle), Incircle (incenter), Centroid, Orthocenter.
1. Construct a Circumcircle (right triangle). Label the Circumcenter.
2. Construct a Circumcircle (acute triangle). Label the Circumcenter.
3. Construct a Circumcircle (obtuse triangle). Label the Circumcenter.
4. Construct an Incircle (right triangle). Label the Incenter.
5. Construct an Incircle (acute triangle). Label the Incenter.
6. Construct an Incircle (obtuse triangle). Label the Incenter.
7. Find and label the Centroid of the triangle (right triangle).
8. Find and label the Centroid of the triangle (acute triangle).
9. Find and label the Centroid of the triangle (obtuse triangle).
10. Find and label the Orthocenter of the triangle (right)
11. Find and label the Orthocenter of the triangle (acute)
12. Find and label the Orthocenter of the triangle (obtuse).
B. General Questions involving points of concurrency:
a. Which points of concurrency are always inside the triangle?
b. Which points of concurrency may fall outside the triangle? In what types of triangles
do these points fall outside?
c. In a right triangle the _____________ falls on the vertex of the right angle.
d. In a right triangle the _____________ falls on the midpoint of the hypothenuse.
e. Which point of concurrency divides each median into a specific ratio? What is that
ratio?
f. Which point of concurrency is the center of balance?
g. Which point of concurrency is equidistant from each vertex?
h. Which point of concurrency is equidistant from each side of a triangle?
i. Which point of concurrency is the center of the largest circle that can be drawn inside
a triangle?
j. Which point of concurrency is the center of a circle that can be drawn touching each
vertex of a triangle?
C. Algebra involving the Centroid:
a. Point G is the Centroid of triangle ABC. AD=10, AG= 12, GF=5, and CD=24. Find
the length of the given segment.
B
̅̅̅̅
𝐵𝐷 =
̅̅̅̅ =
𝐴𝐵
D
̅̅̅̅
𝐸𝐺 =
E
G
̅̅̅̅
𝐴𝐸 =
̅̅̅̅
𝐶𝐺 =
̅̅̅̅
𝐷𝐺 =
̅̅̅̅
𝐵𝐺 =
A
F
C
b. Point G is the Centroid of triangle ABC.
𝐴𝐷 = 2𝑥 + 4, 𝐷𝐵 = 4𝑥 − 6, 𝐵𝐸 = 5𝑥 − 8. Find the length of EC.
B
D
E
G
A
F
c. Find the coordinates of the centroid of a triangle with the given vertices:
A (5, 1) B (6, -4) C ( -2, -3)
d. Find the coordinates of the center of balance of the triangle below:
C
D. Applying the points of concurrency: Hint draw a picture for each one!
a. You want to live in a place that is equal walking distance from the three most
important places: school, your favorite pizza place, and the library. How would you
determine of the location of the place you should live?
b. You want to hang a triangular shaped tray in your room. You will attach a piece of
string to one point on the tray and suspend it from the ceiling. How would you
determine the point where the tray will balance, so that it is parallel to the ground?
c. You want to put the largest possible circular play area in your triangular-shaped
closet for your dog to play in. How would you determine the location of the play
area’s center?
E. Midsegment:
1.
3. In diagram below of
a. Find the perimeter of
2.
,
is a midsegment of
, DE=14, AB= 20, and BC= 26.
.
b. If F is the midpoint of AC, find the perimeter of triangle DEF.
4. On the axes below, graph and label triangle with vertices at D(-3,2) , E(3, 4), and F(1, -6).
a. Find the coordinates of G, the midpoint of ̅̅̅̅
𝐸𝐹
Coordinates of G (
,
)
b. Find the coordinates of H, the midpoint of ̅̅̅̅
𝐷𝐸
Coordinates of H (
,
)
c. Graph and label G and H on your graph.
1
d. Verify that 𝐺𝐻 = 2 𝐷𝐹.
e. Justify the statement 𝐺𝐻 ∥ 𝐷𝐹.