Name:________________________________________________________________
Date:___________________________ Period:_____
Chapter 3 – Constructions Review
Constructions to know:
 Medians
Centers to know:
 Centroid
 Angle Bisectors
 Incenter (with circle!)
Algebra to know:
 1-2-3 Rule
 Coordinates on a plane (average)
 All radii are equal in measure
 Perpendicular Bisectors
 Circumcenter (with circle!)
 All radii are equal in measure
 Altitudes
 Orthocenter
 Congruent Segment
 Congruent Angle
 Equilateral Triangle
 300 Angle
 Parallel Lines
Directions: Answer the following questions completely. Justify your calculations for each question.
1. The intersection of the perpendicular bisectors is called the: _______________________________________
2. The intersection of the angle bisectors is called the: _______________________________________
3. The intersection of the altitudes of a triangle is called the: _______________________________________
4. The intersection of the medians of a triangle is called the: _______________________________________
5. What is the center of the circle that is inscribed inside a triangle? _______________________________________
6. What is the center of the circle that is circumscribed around a triangle? _______________________________________
7. In the diagram below of ΔABC, medians AD, BE, and CF intersect at G.
If CF = 24, what is the length of FG?
(1) 8
(2) 10
(3) 12
(4) 16
8. In the diagram of ΔABC below, Jose found the centroid P by constructing the three medians. He measured CF and
found it to be 6 inches.
If PF = x, which equation can be used to find x?
(1) x + x = 6
(2) 3x + 2x = 6
(3) 2x + x = 6
2
(4) x + 3x = 6
9. In triangle ABC, medians AD , BE , and CF
are concurrent at point P. If AP = 8, find the length
of median AD .
10. P is the incenter of ∆𝑋𝑌𝑍. If m<SZP = 7x + 7,
and m<SZT = 16x + 4, find the value of x and
m<SZT.
11. In triangle ABC, medians AD , BE , and CF
are concurrent at point P. If AP=7x+1 and DP = 4x-2,
then what is the value of x?
12. The circumcenter of ΔABC is point P. If AP=x+2y,
BP = 40, and CP = x + 4, find x and y.
13. Given triangle ABC with coordinates A(3,10),
B(-6,2), and C(-9,3). What are the coordinates of the
centroid of triangle ABC?
14. In triangle ABC, medians AD and BE intersect
15. The perpendicular bisectors of triangle ABC
16. Given triangle ABC with coordinates A (3, -1),
B (9,5), and C (-3, 2), find the coordinates
of the centroid of triangle ABC.
17. In triangle ABC, medians AD , BE , and CF are
concurrent at point P. If FP=x+1 and FC = 6x-12, then
what is the value of x?
18. P is the incenter of ∆𝑋𝑌𝑍. If m<SXP = 7x,
intersect at point P. If AP=20, BP=2x+4, and
CP=3y-7, what is the value of x and y?
at P. If BE = 8𝑥 and BP = 2𝑥 + 10, find the length
of BE.
and m<RXP = 2x+50, find the value of x
and m<SXP.
Directions: Answer the following questions completely.
⃡ parallel to 𝑅𝑄
⃡ through point P.
19. The diagram below illustrates the construction of 𝑃𝑆
Which statement justifies this construction?
(1) 𝑚 < 1 = 𝑚 < 2
(2) 𝑚 < 1 = 𝑚 < 3
(3) ̅̅̅̅
𝑃𝑅 ≅ ̅̅̅̅
𝑅𝑄
̅̅̅̅
(4) 𝑃𝑆 ≅ ̅̅̅̅
𝑅𝑄
20. The diagram shows the construction of the bisector of < 𝐴𝐵𝐶.
Which statement is not true?
1
(1) 𝑚 < 𝐸𝐵𝐹 = 2 𝑚 < 𝐴𝐵𝐶
(2) 𝑚 < 𝐸𝐵𝐹 = 𝑚 < 𝐴𝐵𝐶
1
(3) 𝑚 < 𝐷𝐵𝐹 = 𝑚 < 𝐴𝐵𝐶
2
(4) 𝑚 < 𝐷𝐵𝐹 = 𝑚 < 𝐸𝐵𝐹
̅̅̅̅.
21. The diagram below shows the construction of the perpendicular bisector of 𝐴𝐵
Which statement is not true?
(1) 𝐴𝐶 ≅ 𝐶𝐵
1
(2) 𝐶𝐵 = 2 𝐴𝐵
(3) 𝐴𝐶 = 2𝐴𝐵
(4) 𝐴𝐶 + 𝐶𝐵 = 𝐴𝐵
22. Which geometric principle is used to justify the construction below?
(1) A line perpendicular to one of two parallel lines is perpendicular to the other.
(2) Two lines are perpendicular if they intersect to form congruent adjacent angles.
(3) When two lines are intersected by a transversal and alternate interior angles are congruent, the lines
are parallel.
(4) When two lines are intersected by a transversal and the corresponding angles are congruent, the lines
are parallel.
23. Which diagram shows the construction of the perpendicular bisector of ̅̅̅̅
𝐴𝐵 ?
24. Which diagram represents a correct construction of equilateral ∆𝐴𝐵𝐶, given side ̅̅̅̅
𝐴𝐵.
Directions: Answer the following questions completely using a compass and a straightedge.
Remember to leave all construction marks.
25. Construct an angle congruent to < 𝐴𝐵𝐶. Label this new angle as < 𝐷𝐸𝐹.
26. Construct an angle bisector of < 𝐴𝐵𝐶.
27. Construct an angle bisector of < 𝐵𝐴𝐶.
28. On theline segment below, construct equilateral
triangle ABC.
29. Triangle ABC is shown. Is this an equiateral
Triangle? Justify your answer.
30. Using the equilateral triangle below, construct
a 30 degree angle.
31. Construct a perpendicular bisector.
32. For the following, construct a line perpendicular through the given points.
33. Construct a line parallel through the given point.
34. There are two boys kicking a soccer ball at a field. They are both equidistant from each other. Their friend
wants to join. Where would the third boy have to stand in order for them to all be equidistant from each other.
Label this 𝐵3 .
35. Mrs. Fields lives at home. She has a large, rectangular shaped backyard. Mrs. Fields runs both a dog-sitting
business and a daycare service. She wants to install a fence that is equidistant from the doghouse and the
playground, to keep her businesses separate. Use your construction tools to determine where the fence should go
on the plot of land.
**If you want to practice more constructions, re-draw the diagrams
above in any size you want and you can practice this way**