Chapter 5 Review
Honors Geometry
Mathematician ___________________
Always, sometimes, or never
1. The shortest segment from a point to a line is the perpendicular segment.
2. When the altitude, angle bisector, and median are drawn from the same vertex of a triangle, the altitude is
longer than the other two.
3. Each side of a triangle is greater than the difference of the other two sides.
In ∆ABC, mA  70 . Side BC is the longest side of the triangle.
The incenter is equidistant from the sides of the triangle.
A median of a triangle bisects one of the angles.
If one altitude of a triangle is in the triangle’s exterior, then a second altitude is also in the triangle’s
exterior.
8. The centroid of a triangle lies in its exterior.
4.
5.
6.
7.
Answer each of the following:
9. ∆ABC is an isosceles triangle with AC = BC, and one of the angle is obtuse. The longest side of the triangle
is ____.
10. Two sides of a triangle have lengths 4 and 9. Find the range for the possible lengths for the 3 rd side
11. In ∆ABC, The exterior angle at A is 120 degrees. If mB  mC , then the
longest side of the triangle is _____
12. The centroid of a triangle is 3 cm from one vertex, 4 cm from a second vertex,
and 5 cm from a third vertex. Find the sum of the lengths of the three medians
of the triangle.
13. If the centroid and the circumcenter of a triangle are located at the same point, then
the triangle is a) equilateral b) isosceles c) scalene d) a, b or c d) not possible
14. If C is the centroid of ∆ATP and located on
AS , find the length of AS .
A
C (3, 4)
T (-1, -2)
S
P (5, -2)
15. Stained glass artist Sue wishes to inscribe a circle in a triangular portion of her latest
design. Which point of concurrency in the triangular section of her design does Sue
need to locate?
16. Ryan has built a home for his pet hamsters. It is in the shape of a triangular prism. He
wants to cut out the largest possible circular entrance from one of the bases. Which
point of concurrency in the triangular base does Ryan need to locate?
17. Emily is redesigning her kitchen. She wishes to put an island at a location equidistant
from the refrigerator, oven & sink. Which pt of concurrency does Emily need to locate?
18. In triangle ABC, AC > BC > AB List the three angles in order of size from largest to
smallest.
19. Given: ∆FJH is isosceles, with base JH. K is the midpoint of FJ and G is the midpoint of
FH. FK = 2x + 3, GH = 5x-9, and JH = 4x. Find the perimeter of ∆FJH.
20. Given ∆ABC. A (4, 8), B (2, 1), C (12, 3). Find the coordinates of the point where the
median from A intersects BC.
Name the segments illustrated.
21.
22.
23.
24.
25.
67 67
43
43
26. Write the equation of the perpendicular bisector of the segment with endpoints A (2, -6) B (4, 0).
27. Find the circumcenter of a triangle with vertices A(0, 6) B(8, 0) C (0, 0)
28. Find the orthocenter of a triangle with vertices A(1, 2) B(6, 2) C (1, -8)
29. The vertices of ΔDEF are D(-4, -7) E (2, 5) F (10, -3). V is the midpoint of DE and W is the midpoint of
EF. Prove VW is a midsegment.
30. GJ is a midsegment of ΔDEF and HK is a midsegment of ΔGFJ. What is the length of HK?
D
2n -1
G
H
2n + 1
7cm
F
K
J
E
31. What is the longest side in the picture?
P
59 58
M
60
62
61
O
60
N
Solutions: 1. Always 2. Never 3. Always
4. Sometimes 5. Always 6. Sometimes 7. Always 8. Never 9. AB
10. 5 < x < 13 11. AC 12. 18cm 13. A 14. 3 37 15. incenter 16. incenter 17. circumcenter 18. angles B, A, C 19. 60 20. (7, 2)
21. Medians/Centroid 22. angle bisectors/Incenter 23. Altitudes/Orthocenter 24. Perpendicular Bisectors/Circumcenter
25. Midsegment 26. y =
1
x -2 27. (4,3) 28. (1,2)
29. Show VW
DF are parallel by calculating the slope of each. Then calculate
3
the distance of VW and DF and show VW = half of DF.
Slope of each =
2
VW =
DF =
53
2 53
7
The finished statement looks like this: slope of VW =
2
7
Therefore, VW
30. 4 cm 31. MP (no, this is not a mistake)
slope of DF =
2
7
DF
VW =
1
DF
=
1
53 =
( 2 53 )
2
2
Therefore, VW is a midsegment.
53
53
√