Name:________________________________________________________________ Date:___________________________ Period:_____Chapter 3 – Constructions Review

Constructions to know: Centers to know: Algebra to know: Medians Centroid 1-2-3 Rule

Coordinates on a plane (average) Angle Bisectors Incenter (with circle!) 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 ABC, medians AD, BE, and CF intersect at G.Δ

If CF = 24, what is the length of FG?(1) 8 (2) 10 (3) 12 (4) 16

8. In the diagram of ABC below, Jose found the centroid P by constructing the three medians. He measured CF andΔ 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 (4) x + 23x = 6

9. In triangle ABC, medians , , and 10. P is the incenter of ∆ XYZ. If m<SZP = 7x + 7, are concurrent at point P. If AP = 8, find the length and m<SZT = 16x + 4, find the value of x and

of median . m<SZT.

11. In triangle ABC, medians , , and 12. The circumcenter of ABC is point P. If AP=x+2y,Δare concurrent at point P. If AP=7x+1 and DP = 4x-2, BP = 40, and CP = x + 4, find x and y. then what is the value of x?

13. Given triangle ABC with coordinates A(3,10), 14. In triangle ABC, medians AD and BE intersect B(-6,2), and C(-9,3). What are the coordinates of the at P. If BE = 8 x and BP = 2 x+10, find the lengthcentroid of triangle ABC? of BE.

15. The perpendicular bisectors of triangle ABC 16. Given triangle ABC with coordinates A (3, -1),intersect at point P. If AP=20, BP=2x+4, and B (9,5), and C (-3, 2), find the coordinatesCP=3y-7, what is the value of x and y? of the centroid of triangle ABC.

17. In triangle ABC, medians , , and are 18. P is the incenter of ∆ XYZ. If m<SXP = 7x,concurrent at point P. If FP=x+1 and FC = 6x-12, then and m<RXP = 2x+50, find the value of xwhat is the value of x? and m<SXP.

Directions: Answer the following questions completely.

19. The diagram below illustrates the construction of PS parallel to RQ through point P.

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) m<EBF=12m<ABC

(2) m<EBF=m<ABC

(3) m<DBF=12m<ABC

(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

(2) CB=12AB

(3) AC=2 AB(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 ¿≝¿.

26. Construct an angle bisector of ¿ ABC. 27. Construct an angle bisector of ¿BAC.

28. On theline segment below, construct equilateral 29. Triangle ABC is shown. Is this an equiateraltriangle ABC. Triangle? Justify your answer.

30. Using the equilateral triangle below, construct 31. Construct a perpendicular bisector.a 30 degree angle.

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**