Midsegments of Triangles Date: Per - CUSD Math

Math 2 Unit 4 Worksheet 1A

Math 2 Unit 4 Worksheet 1A Name: Midsegments of Triangles Date: Per: [1-2] Prove the midsegment triangle theorem using coordinate geometry.

1. Using ∆ 𝐽𝐽𝐽𝐽𝐽𝐽 answer the following questions.

a. Find the coordinate of the midpoint of 𝐽𝐽𝐽𝐽� . Label it point 𝑀𝑀. *Midpoint formula 𝒙𝒙𝟏𝟏 +𝒙𝒙𝟐𝟐

𝟐𝟐, 𝒚𝒚𝟏𝟏+𝒚𝒚𝟐𝟐

𝟐𝟐

b. Find the coordinate of the midpoint of 𝐽𝐽𝐽𝐽��. Label it point 𝑁𝑁.

c. Find the slopes of 𝑀𝑀𝑁𝑁�� and 𝐽𝐽𝐽𝐽�� using the slope formula 𝒚𝒚𝟐𝟐− 𝒚𝒚𝟏𝟏𝒙𝒙𝟐𝟐−𝒙𝒙𝟏𝟏

. Justify mathematically why they are parallel.

d. Using the distance formula, �(𝑥𝑥2 − 𝑥𝑥1)2 + (𝑦𝑦2 − 𝑦𝑦1)2 find the length of 𝑀𝑀𝑁𝑁�� and the length of 𝐽𝐽𝐽𝐽��.

e. The length of 𝑀𝑀𝑁𝑁�� is _____________ the length of 𝐽𝐽𝐽𝐽��. From your work in part d, explain mathematically why this is true.

𝐽𝐽 (0, 0)

𝐽𝐽 (4, 2)

Theorem: The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.

𝐽𝐽 (2, 6)

Math 2 Unit 4 Worksheet 1A

2. The coordinates of the vertices of a triangle are: 𝐴𝐴 ( − 2, 3 ) 𝐵𝐵 ( 6, 1 ) 𝐶𝐶 ( 4, 5 ) Plot and label the points on the graph.

a. Find the coordinate of the midpoint of 𝐴𝐴𝐶𝐶��. Label it point 𝐷𝐷.

b. Find the coordinate of the midpoint of 𝐶𝐶𝐵𝐵��.

Label it point 𝐸𝐸.

c. Mathematically show that 𝐷𝐷𝐸𝐸�� is parallel to 𝐴𝐴𝐵𝐵��.

d. Mathematically show that the length of 𝐷𝐷𝐸𝐸�� is half the length of 𝐴𝐴𝐵𝐵��.

3. 𝐵𝐵 is the midpoint of 𝐴𝐴𝐶𝐶��, D is the midpoint of 𝐴𝐴𝐸𝐸�� . 𝑚𝑚∠ 𝐴𝐴𝐷𝐷𝐵𝐵 = 70ᵒ, 𝑚𝑚∠ 𝐶𝐶 = 60ᵒ, 𝐵𝐵𝐷𝐷 = 12

a. Find 𝐶𝐶𝐸𝐸

b. Find 𝑚𝑚∠ 𝐸𝐸 c. Find 𝑚𝑚∠ 𝐴𝐴𝐵𝐵𝐷𝐷 d. Find 𝑚𝑚∠ 𝐴𝐴

𝐸𝐸 𝐷𝐷

𝐶𝐶

𝐵𝐵

𝐴𝐴

Math 2 Unit 4 Worksheet 1B

Math 2 Unit 4 Worksheet 1B Name: Midsegments of Triangles Date: Per: [1-4] Use the diagram. 𝑋𝑋 is the midpoint of 𝑈𝑈𝑈𝑈. 𝑌𝑌 is the midpoint of 𝑈𝑈𝑈𝑈.

1. If 𝑚𝑚∠𝑈𝑈𝑋𝑋𝑌𝑌 = 60°, find m∠V 2. If m∠W = 45°, find 𝑚𝑚∠𝑈𝑈𝑌𝑌𝑋𝑋

3. If 𝑋𝑋𝑌𝑌 = 50, find 𝑈𝑈𝑈𝑈 4. If 𝑈𝑈𝑈𝑈 = 110, find 𝑋𝑋𝑌𝑌 [5-6] Use the diagram. 𝐼𝐼𝐼𝐼� is a midsegment of 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥. 𝐼𝐼𝐼𝐼 = 7, 𝛥𝛥𝛥𝛥 = 10, and 𝛥𝛥𝛥𝛥 = 13. Find the perimeter of each triangle.

5. 𝛥𝛥𝐼𝐼𝐼𝐼𝛥𝛥 6. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 [7-9] Find the value of the x.

7. 𝑥𝑥 = 8. 𝑥𝑥 =

9. 𝑥𝑥 =

𝑈𝑈

𝑋𝑋 𝑌𝑌

𝑈𝑈 𝑈𝑈

𝛥𝛥

𝐼𝐼 𝐼𝐼

𝛥𝛥 𝛥𝛥

Math 2 Unit 4 Worksheet 1B

[10-12] Use the given diagram to solve.

10. If 𝐷𝐷𝛥𝛥 = 24, 𝐵𝐵𝐵𝐵 = 6, and 𝐷𝐷𝐵𝐵 = 8, find the perimeter of 𝛥𝛥𝛥𝛥𝐷𝐷𝛥𝛥.

11. If 𝐵𝐵𝐵𝐵 = 2𝑥𝑥 + 6 and 𝐷𝐷𝛥𝛥 = 5𝑥𝑥 + 9, find 𝐷𝐷𝛥𝛥.

12. If 𝐵𝐵𝐵𝐵 = 3𝑥𝑥 – 1 and 𝛥𝛥𝐷𝐷 = 5𝑥𝑥 + 7, find 𝐵𝐵𝐵𝐵. [13-14] Find the value of the variable(s).

13. 𝑥𝑥 = 14. 𝑥𝑥 = 𝑦𝑦 =

Math 2 Unit 4 Worksheet 2

Math 2 Unit 4 Worksheet 2 Name: Perpendicular Bisectors Date: Per:

[1-5] Complete the below chart. 1.

𝐶𝐶𝐶𝐶�� is a ______________________________ of 𝐴𝐴𝐴𝐴��

2.

Point 𝐾𝐾 is the _____________________________ of the triangle.

3. Distance Formula

4. Midpoint Formula

5. Slope Formula

6. When 3 lines meet all in one point they are called ________________________. The point where they meet is

called the point of ________________________.

7. Every point on the _______________________ of a segment is equidistant from the ______________________

of that segment.

[8-11] Use the figure to the right to answer the set of questions.

8. a) What is the relationship between 𝐴𝐴𝐶𝐶 and 𝐴𝐴𝐶𝐶 ? b) What is the value of 𝑥𝑥? c) Find 𝐴𝐴𝐴𝐴. d) Find 𝐴𝐴𝐶𝐶.

9. a) From the information given in the figure, how is 𝑇𝑇𝑇𝑇 related to 𝑆𝑆𝑆𝑆 ? b) Find 𝑇𝑇𝑆𝑆. c) Find 𝑆𝑆𝑇𝑇. d) Find 𝑆𝑆𝑆𝑆.

10. a) 𝑀𝑀𝑀𝑀 is the perpendicular bisector of _________ b) Find 𝑀𝑀𝑀𝑀. c) Find 𝑁𝑁𝑀𝑀. d) Find 𝑁𝑁𝑀𝑀.


11. a) _________ is the perpendicular bisector of segment _________ b) What are the lengths of 𝐸𝐸𝐸𝐸 and 𝐸𝐸𝐸𝐸 ? c) Find the value of 𝑦𝑦. d) Find 𝑚𝑚∠𝐺𝐺𝐸𝐸𝐸𝐸 and 𝑚𝑚∠𝐺𝐺𝐸𝐸𝐸𝐸.

12. Make a two-column proof.

Given: 𝐶𝐶𝑀𝑀�⃖��⃗ is a ⊥ bisector of 𝐴𝐴𝐴𝐴.�� Prove: 𝐴𝐴𝐶𝐶�� ≅ 𝐴𝐴𝐶𝐶��

13. Find the circumcenter of the triangle.

14. Where should the farmer place the hay bale 15. Where should the fire station be placed so that it is

so that it is equidistant from the three gates? Equidistant from the grocery store, the hospital, and the police station?

Statement Reason

𝐴𝐴 𝐴𝐴

𝐶𝐶

𝑀𝑀


Math 2 Unit 4 Worksheet 3 Name: Angle Bisectors Date: Per:

1. a) Draw the angle bisector of ∠𝐴𝐴. 2. a) Draw the perpendicular bisector of 𝐷𝐷𝐷𝐷��.

b) Mark the diagram showing congruent angles. b) Mark your diagram showing right angles and congruent segments.

3. If a point is on the ________________________ of an angle, then the point is ________________________ from the side of the angle.

[4-5] Find the value of 𝑥𝑥.

4. 𝑥𝑥 = _________ 5. 𝑥𝑥 = _________ [6-7] Find the indicated values of the variables and measures.

6. 𝑥𝑥 = _________ 7. 𝑥𝑥 = _________

𝐵𝐵𝐴𝐴 = ________ ∠𝐵𝐵𝐴𝐴𝐵𝐵 = ________

𝐷𝐷𝐴𝐴 =________ ∠𝐷𝐷𝐴𝐴𝐵𝐵 =________

[8-13] Use the figure to the right to answer the following questions.

8. How far is 𝑀𝑀 from 𝐾𝐾𝐾𝐾�� ?

9. How far is 𝑀𝑀 from 𝐽𝐽𝐾𝐾�� ?

10. How is 𝐾𝐾𝑀𝑀�� related to ∠𝐽𝐽𝐾𝐾𝐾𝐾?

11. Find the value of 𝑥𝑥.

12. Find ∠𝑀𝑀𝐾𝐾𝐽𝐽.

13. Find ∠𝐽𝐽𝑀𝑀𝐾𝐾 and ∠𝐾𝐾𝑀𝑀𝐾𝐾.

𝐷𝐷 𝐷𝐷

𝐸𝐸

𝐴𝐴 𝐵𝐵

𝐵𝐵


14. Is 𝐴𝐴 on the angle bisector of ∠𝑋𝑋𝑋𝑋𝑋𝑋? Explain.

[15-16] Use the figure on the right to answer the following questions.

15. a) Point of concurrency is called ____________________.

Incenter / Circumcenter

b) This is the center of the _________________________. Inscribed Circle / Circumscribed Circle c) Use a compass to sketch the circle.

16. a) Point of concurrency is called ____________________. Incenter / Circumcenter

b) This is the center of the _________________________. Inscribed Circle / Circumscribed Circle c) Use a compass to sketch the circle.


Math 2 Unit 4 Worksheet 4 Name: Medians & Altitudes Date: Per:

[1-10] Complete the below chart. 1.

𝐶𝐶𝐶𝐶�� is a ______________________________ of 𝐴𝐴𝐴𝐴��

2.

Point 𝐾𝐾 is the _____________________________ of the triangle.

3.

Point 𝑃𝑃 is the _____________________________ of the triangle.

4.

The segment shown is an _____________________________.

5.

Point 𝑄𝑄 is the _____________________________ of the triangle.

6.

The segment shown is a _____________________________.

7.

Point 𝑊𝑊 is the _____________________________ of the triangle.

8. Distance Formula

9. Midpoint Formula

10. Slope Formula

[11-13] Is 𝑀𝑀𝑀𝑀�� a median, an altitude, or neither? Explain.

11. _______________________ 12. _______________________ 13. _______________________


[14-17] Is 𝐴𝐴𝐴𝐴�� a median, an altitude, or neither? Explain.

14. ____________________________ 15. ____________________________

16. ____________________________ 17. ____________________________ [18-21] Name each Segment.

18. A median in ∆𝑆𝑆𝑆𝑆𝑆𝑆 ________________

19. An altitude in ∆𝑆𝑆𝑆𝑆𝑆𝑆 _______________

20. A median in ∆𝑆𝑆𝐴𝐴𝑆𝑆 ________________

21. An altitude in ∆𝐶𝐶𝐴𝐴𝑆𝑆 _______________

[22-24] In ∆𝑋𝑋𝑋𝑋𝑋𝑋, 𝐴𝐴 is the centroid.

22. If 𝐶𝐶𝑋𝑋 = 12, find 𝑋𝑋𝐴𝐴, 𝐴𝐴𝐶𝐶, and describe the relationship between 𝑋𝑋𝐴𝐴 and 𝐶𝐶𝑋𝑋. 𝑋𝑋𝐴𝐴 = ___________ 𝐴𝐴𝐶𝐶 = ___________ 𝑋𝑋𝐴𝐴 = _______ of 𝐶𝐶𝑋𝑋

23. If 𝐴𝐴𝐴𝐴 = 6, find 𝐴𝐴𝑋𝑋 and 𝐴𝐴𝑋𝑋. 24. If 𝐴𝐴𝐶𝐶 = 3, find 𝐶𝐶𝑋𝑋 and 𝐴𝐴𝑋𝑋.

𝐴𝐴𝑋𝑋 = ___________ 𝐶𝐶𝑋𝑋 = ___________ 𝐴𝐴𝑋𝑋 = ___________ 𝐴𝐴𝑋𝑋 = ___________

25. 𝑄𝑄 is the centroid of ∆𝐽𝐽𝐾𝐾𝐽𝐽. 𝑃𝑃𝐾𝐾 = 9𝑥𝑥 + 21𝑦𝑦. Write an expression to represent 𝑃𝑃𝑄𝑄 and 𝑄𝑄𝐾𝐾.


Review [26-30] Find the value of 𝑥𝑥.

26. 𝑥𝑥 = _______ 27. 𝑥𝑥 = _______ 28. 𝑥𝑥 = _______

29. 𝑥𝑥 = _______ 30. 𝑥𝑥 = _______ [31-33] 𝑋𝑋 is the midpoint of 𝑀𝑀𝑀𝑀��. 𝑋𝑋 is the midpoint of 𝑂𝑂𝑀𝑀��. 𝑋𝑋 is the midpoint of 𝑀𝑀𝑂𝑂��

31. Find 𝑋𝑋𝑋𝑋.

32. If 𝑋𝑋𝑋𝑋 = 10, find 𝑀𝑀𝑂𝑂.

33. If 𝑚𝑚∠𝑀𝑀 is 64°, find 𝑚𝑚∠𝑋𝑋𝑋𝑋𝑋𝑋. [34-35] Use the diagram to answer the questions.

34. What is the distance across the lake?

35. Is it a shorter distance from 𝐴𝐴 to 𝐴𝐴 or from 𝐴𝐴 to 𝐶𝐶? Explain.


Math 2 Unit 4 Worksheet 6 Name: Inequalities in One and Two Triangles Date: Per: [1-6] List the sides and the angles in order from smallest to largest.

1. 2. Sides: ____________________ Sides: ____________________ Angles: ___________________ Angles: ___________________ 3. 4.

Sides: ____________________ Sides: ____________________ Angles: ___________________ Angles: ___________________

5. 6.

Sides: ____________________ Sides: ____________________ Angles: ___________________ Angles: ___________________

7. Multiple Choice: In ∆𝑅𝑅𝑅𝑅𝑅𝑅, Which is a possible side length for 𝑅𝑅𝑅𝑅?

[8-11] Is it possible to construct a triangle with the given side lengths? If not, explain why not.

8. 6, 7, 11 9. 3, 6, 9 10. 28, 34, 39 11. 35, 120, 125

𝐶𝐶 62°

𝐵𝐵

67°

𝐴𝐴 51°

6

𝑅𝑅 𝑅𝑅

𝑅𝑅

10

9

112° 32°

36°

𝑋𝑋

𝑌𝑌 𝑍𝑍

127°

29° 𝑀𝑀

𝑁𝑁

𝑃𝑃

𝐹𝐹 𝐺𝐺

𝐷𝐷

33°

28

𝐾𝐾

𝐿𝐿 𝐽𝐽 25

13

8

𝑅𝑅 𝑅𝑅

𝑅𝑅

65° 56° A 7 B 8

C 9 D Cannot be determined


12. Multiple Choice: Which group of side lengths can be used to construct a triangle? a) 3yd., 4ft., 5yd. b) 3 yd., 5 ft., 8 ft. c) 11 in., 16 in., 27 in. d) 2ft., 11 in., 12 in.

[13-18] Describe the possible lengths of the third side of the triangle given the lengths of the other 2 sides.

13. 5 inches, 12 inches 14. 3 meters, 4 meters 15. 12 feet, 18 feet

16. 10 yards, 23 yards 17. 2 feet, 40 inches 18. 25 meters, 25 meters [19-21] Determine which segment is the shortest in each diagram.

19. 20. 21.

22. Describe your process for determining which segment is the shortest when you have two triangles with a shared side. ______________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________

Math 2 Unit 4 Review Worksheet

Math 2 Unit 4 Name: Review Worksheet Date: Per: [1-2] Use the figure to the right to answer the questions.

1. 𝐷𝐷𝐷𝐷�� is a midsegment of ∆𝐴𝐴𝐴𝐴𝐴𝐴. If 𝐷𝐷𝐷𝐷 = 𝑥𝑥 + 5 and 𝐴𝐴𝐴𝐴 = 3𝑥𝑥 − 1, find the value of 𝑥𝑥.

2. 𝐷𝐷𝐷𝐷�� is a midsegment of ∆𝐴𝐴𝐴𝐴𝐴𝐴. If 𝐴𝐴𝐷𝐷 = 15 and 𝐴𝐴𝐷𝐷 = 14, find the value of 𝐴𝐴𝐴𝐴.

3. Multiple Choice: Which group of side lengths can be used to construct a triangle? Show work. a) 5 in, 8 in, 15 in b) 10 cm, 12 cm, 20 cm c) 15 in, 10 in, 25 in d) 18 cm, 2 cm, 15 cm

4. List the order from shortest to longest side in triangle ∆𝐴𝐴𝐴𝐴𝐴𝐴.

5. The lengths of two sides of a triangle are 41 and 23. What are the possible lengths of the third side?

6. Match the line with the correct description.

1. Perpendicular Bisector______ a. From a midpoint to the opposite vertex

2. Angle Bisector_______ b. Bisects an angle

3. Median_______ c. Perpendicular to a side and goes through the opposite vertex

4. Altitude________ d. Perpendicular line through the midpoint of a side

7. 𝐴𝐴𝐷𝐷�� is a median in ∆𝐴𝐴𝐴𝐴𝐵𝐵. Are the below statements True or False?

a. 𝐴𝐴𝐴𝐴�� ≅ 𝐴𝐴𝐵𝐵�� ______

b. ∠𝐴𝐴 ≅ ∠𝐵𝐵 ______

c. 𝐴𝐴𝐷𝐷�� ≅ 𝐷𝐷𝐵𝐵�� ______

d. ∠𝐴𝐴𝐴𝐴𝐷𝐷 ≅ ∠𝐵𝐵𝐴𝐴𝐷𝐷______

B

C

A

𝟓𝟓𝟓𝟓°

𝟖𝟖𝟓𝟓°

𝟒𝟒𝟒𝟒°

C B

A

x + 5 D E

3x -1

B E

A

N


[8-11] Use the given triangles to draw a segment showing the following:

8. Altitude 9. Angle Bisector

10. Perpendicular Bisector 11. Median

12. What is the name of the intersecting lines in the triangle? What is the name of the point of concurrency?

𝐴𝐴.___________________&__________________ 𝐴𝐴.___________________&__________________ 𝐴𝐴.___________________&__________________ 𝐷𝐷.___________________&__________________

13. Find 𝐴𝐴𝐷𝐷�� 14. Find 𝑥𝑥, 𝐿𝐿𝐿𝐿 and 𝐷𝐷𝐿𝐿 𝐴𝐴𝐷𝐷�� = _________ 𝑥𝑥 = _________

𝐿𝐿𝐿𝐿 = _________

𝐷𝐷𝐿𝐿 = _________

Z Y

X

Z Y

X

Z Y

X

Z Y

X

𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐷𝐷

𝐿𝐿 𝐷𝐷

𝐿𝐿

𝑈𝑈

10𝑥𝑥 + 7 8𝑥𝑥 + 24


15. 𝑋𝑋 is the midpoint of 𝑀𝑀𝐵𝐵��. 𝑌𝑌 is the midpoint of 𝑂𝑂𝐵𝐵��. 𝑍𝑍 is the midpoint of 𝑀𝑀𝑂𝑂�� Find 𝑋𝑋𝑍𝑍 = _________ If 𝑋𝑋𝑌𝑌 = 10, find 𝑀𝑀𝑂𝑂. 𝑀𝑀𝑂𝑂 = _________

If 𝐵𝐵𝑂𝑂 = 18, and 𝑀𝑀𝐵𝐵 = 13 find 𝑀𝑀𝑋𝑋. 𝑀𝑀𝑋𝑋 = _________

16. Using the figure to the right, find 𝑥𝑥, 𝐿𝐿𝑇𝑇 and 𝑇𝑇𝑍𝑍 𝑥𝑥 = _________

𝐿𝐿𝑇𝑇 = _________

𝑇𝑇𝑍𝑍 = _________

17. Using the figure to the right, find the value of x.

𝑥𝑥 = _________

18. Which is the smallest angle in ΔMNO? 19. Order the sides of ∆𝐷𝐷𝐷𝐷𝐷𝐷 from shortest to longest.

20. Which segment in each figure would be the shortest?

a) __________________ b) __________________ c) __________________

23° 87° 25°

82°

𝐴𝐴 𝐷𝐷

𝐴𝐴 𝐴𝐴


21. Two sides of a triangle are 5 inches and 10 inches long. What is the range of possible lengths for the third side?

22. The coordinates of the vertices of a triangle are 𝐴𝐴 ( − 2, 3 ), 𝐴𝐴 ( 0,−5 ), 𝐴𝐴 ( −4,−1 ). a) Plot and label the above points on the graph.

b) Find and label the points on the graph for, the

coordinates of D, the midpoint of 𝐴𝐴𝐴𝐴��, coordinates of E, the midpoint of 𝐴𝐴𝐴𝐴��, coordinates of F, the midpoint of 𝐴𝐴𝐴𝐴��

c) Using slopes show that each midsegments is parallel to a side.

d) Using distance formulas show that each midsegment is half the side length.

23. Make a two-column proof.

Given: 𝑅𝑅𝑅𝑅�� ≅ 𝑅𝑅𝐿𝐿�� 𝑅𝑅𝑋𝑋�� is a median Prove: ∠𝑅𝑅𝑅𝑅𝑋𝑋 ≅ ∠𝐿𝐿𝑅𝑅𝑋𝑋

Statement Reason

𝑅𝑅 𝐿𝐿

𝑅𝑅

𝑋𝑋

Midsegments of Triangles Date: Per - CUSD Math

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