1. Construct a perpendicular bisector to the given lines. a) · 2020. 3. 3. · Day 66 Practice Name _____ HighSchoolMathTeachers@2020 Page 13 18. Write the equation to relate RK

Day 66 Bellringer Name ____________________________________

HighSchoolMathTeachers@2020 Page 1

1. Construct a perpendicular bisector to the given lines.

a)

b)



c)

d)



2. Identify two sides that are congruent in the triangle below.

45°

45°

A B

C



Answer Key Day 66

1. a)

b)



c)

d)

2. AC and AB

Day 66 Activity Name ____________________________________


1. Draw a line of length 2𝑖𝑛 and label it AB.

2. Position the compass at end A and extend it to end B, then draw a circle as shown below.

A B



3. Position the compass at the end B and using the same compass width draw a circle as shown

A B



4. Join the intersections of the circles with a straight line as shown. Label the point where the

two lines meet as O.

A B O



5. Measure the length of AO. What do you get?

6. Measure the length of BO. What do you get?

7. Compare the two measurements in 5 and 6 above.



In this activity, students will construct a bisector to the line and measure the resulting portions to

see if they are equal. Students will work in groups of at least three and each group is required to

have a compass, a pencil, a ruler and a plain paper.

Answer Keys

Day 66:

1-4. No response

5. 2 𝑖𝑛

6. 2 𝑖𝑛

7. They are equal

Day 66 Practice Name ____________________________________


Use the diagram below to answer questions 1 and 2.

The point O is a bisector of the line QR. 𝑄𝑂 = 1.4𝑖𝑛

1. What is the length of OR?

2. Find the length of QR

Use the following information to answer questions 3 and 4.

A point S is a bisector to the line AB. The length of AB is 12in.

3. What is the length of SB?

4. What is the length of AS?

5. At what ratio does point S divide line AB?

Line RT divides another line UV in a ratio 1: 1. The two lines intersect at a point O.

The length of OV is 2.4𝑖𝑛. Use this information to answer questions 6 and 7.

6. What is the length of OU?

7. Find the length of the line UV.

The line bisector of line CD intersects it at a point O. Line CD is 8𝑖𝑛 long. Use this information

to answer questions 8-10.

8. What is the length of CO?

O Q R



9. What is the length of OD?

10. Write an equation relating OD and CD.

11. Write an equation relating OD and OC?

Use the following information to answer questions 12-14

A man wanted to erect a security light post exactly in the middle of his rectangular plot. The plot

measured 60ft by 80ft. In order to identify the middle of the plot he drew perpendicular bisectors

to two adjacent sides and erected it at the point of the intersection of the two bisectors.

12. What was the shortest distance from the post to the short side of the plot?

13. What was the shortest distance from the post to the long side of the plot?

14. What was the distance from one vertex of the plot to the post?

A point T divides line RK in the ratio 1:1. Line RK is 24in long.

Use this information to answer questions 15 to 17.

15. Find the length of RT.

16. Find the length of TK.

17. Find the RT:TK



18. Write the equation to relate RK and TK.

A line AB bisects line ST through at a point Q. Line ST 13in long.

Use this information to answer question 19 and 20.

19. What is the length of QT?

20. What is the length of QS?



Answer Keys

Day 66:

1. 1.4𝑖𝑛

2. 2.8𝑖𝑛

3. 6𝑖𝑛

4. 6𝑖𝑛

5. 1: 1

6. 2.4𝑖𝑛

7. 4.8𝑖𝑛

8. 4𝑖𝑛

9. 4𝑖𝑛

10. 𝐶𝐷 = 2𝑂𝐷

11. 𝑂𝐶 = 𝐶𝐷

12. 40 ft

13. 30 ft

14. 50ft

15. 12𝑖𝑛

16. 12𝑖𝑛

17. 2: 1

18. RK=2TK

19. 6.5𝑖𝑛

20. 6.5𝑖𝑛

Day 66 Exit Slip Name ____________________________________


1. Line PQ of length 10in is bisected by a point O. What is the sum of the lengths of the two

portions?



Answer Keys

Day 66:

1. 10𝑖𝑛



Consider ∆KLM (not drawn to scale) shown below where NP = 2.3 in, LN = 3.1 in, NQ = 4.2 in

and KQNP is a parallelogram. Use it to answer the following questions.

(a) Given that KQNP is a parallelogram find the length KP.

(b) Hence find the length KM.

(c) Find the length KQ on parallelogram KQNP

(d) Hence find the length KL.

(e) Compare the length of KL to that of NP. What do you notice?

K L

M

N P

Q

2.3 in

3.1 in 4.2 in



Answer keys Day 67:

(a) KP = 4.2 in.

(b) KM = 8.4 in.

(c) KQ = 2.3 in.

(d) KL = 4.6 in

(e) The length of NP is half that of KL



1. On the plain paper, draw a triangle of suitable size and label it ΔKLM just like the triangle

shown below.

2. Measure the length of KM̅̅̅̅̅ and a hence carefully locate point N, the midpoint of KM̅̅̅̅̅ as shown

below.

3. Similarly, measure the length of ML̅̅ ̅̅ and a hence carefully locate point P, the midpoint of ML̅̅ ̅̅

as shown below.

M

K L

M

K L

N

M

K L

N P



4. Join point N to point P using a ruler as shown below.

5. Measure the lengths of NP̅̅ ̅̅ and KL̅̅̅̅ . What do you notice after comparing these lengths?

6. Now, join point P to point K using a broken line as shown below.

7. Measure ∠NPK and ∠LKP then compare their measures. What main conclusion can be drawn

about the relationship between NP̅̅ ̅̅ and KL̅̅̅̅ ?

M

K L

N P

M

K L

N P



In this activity, students will discover the triangle midpoint theorem by drawing a line joining the

midpoints of any two sides of a triangle. Students can work in groups of three or four. Each

group should have a plain paper, a ruler and a protractor.

Answer keys Day 67:

1. No response

2. Ensure that KN ≅ MN

3. Ensure that MP ≅ LP

4. No response

5. The length of NP̅̅ ̅̅ is half that KL̅̅̅̅

6. No response

7. NP̅̅ ̅̅ ∥ KL̅̅̅̅



Use the figure below to answer questions 1 - 8.

In the figure, S is the midpoint of PQ, ST is parallel to QR, SU is parallel to PR, ∠𝑃𝑆𝑇 = 58°

and ∠𝑃𝑅𝑄 = 58°.

1. Find the measure of ∠𝑆𝑄𝑈

2. Find the measure of ∠𝑄𝑈𝑆

3. Find the measure of ∠𝑃𝑇𝑆

4. Find the measure of ∠𝑇𝑃𝑆

5. Find the measure of ∠𝑄𝑆𝑈

6. Considering the angle measures you have found in questions. State whether Δ𝑃𝑆𝑇 is

congruent to Δ𝑆𝑄𝑈 or not.

P

Q R

S T

U



7. Given that TS ≅ RU, compare the length TS to the length RQ.

8. Considering your answer in question 7 above, is U is the midpoint of QR?

Use the figure below to answer questions 9-12.

In ΔKLM below, N, Q and P are the midpoints of KL, LM and KM respectively and 𝐿𝑁 ∥ 𝑄𝑃.

Show that LQPN is a parallelogram by completing the table below.

Statement Reasons

𝐿𝑁 ∥ 𝑄𝑃 9.

𝑁𝑃 ∥ 𝐿𝑄 10.

11. QP ≅1

2LK but LN ≅

1

2LK

LQ ≅ NP

12.

K

L M

N P

Q



Use the figure below to answer questions 13-20. E and D are the midpoints of AB and AC

respectively, 𝐸𝐷 ≅ 𝐷𝐹 and 𝐶𝐹 ∥ 𝐵𝐸

Given that E and D are the midpoints of AB and AC respectively, give two relationships between

ED and BC in the table below?

13.

14.

Fill in the table below.

Measure of ∠𝑪𝑫𝑭 Reason

15. 16.

Find the measures of the following pairs of angles:

17.∠𝐶𝐵𝐸, hence ∠𝐵𝐸𝐷

18.∠𝐵𝐶𝐹, hence ∠𝐶𝐹𝐷

19. Compare the pairs of angles in question 17 and 18. What do you notice?

20. What type of quadrilateral is most likely to be BCFE according to the angles you have found

questions 17 and 18 above?

A

B C

D E F 59°

61°



Answer keys Day 67:

1. 58°

2. 58°

3. 58°

4. 64°

5. 64°

6. They are congruent

7. TS ≅1

2RQ/ RQ ≅ 2TS

8. Yes

13. ED ≅

1

2BC / BC ≅ 2ED

14. 𝐸𝐷 ∥ 𝐵𝐶

Measure of ∠𝑪𝑫𝑭 Reason

15. 59° 16. Vertical angles / Opposite angles

17. ∠𝐶𝐵𝐸 = 60°, ∠𝐵𝐸𝐷 = 120°

18. ∠𝐵𝐶𝐹 = 120°, ∠𝐶𝐹𝐷 = 60°

19. The angles in each pair are congruent

20. A parallelogram

Statement Reasons

𝐿𝑁 ∥ 𝑄𝑃 9. Triangle midpoint theorem

𝑁𝑃 ∥ 𝐿𝑄 10. Triangle midpoint theorem

11. 𝑳𝑵 ≅ 𝑸𝑷 QP ≅1

2LK but LN ≅

1

2LK

LQ ≅ NP 12. 𝐍𝐏 ≅𝟏

𝟐𝐋𝐌 but 𝐋𝐐 ≅

𝟏

𝟐𝐋𝐌



In the figure below TYUS is a parallelogram in ΔXYZ. YU̅̅ ̅̅ is produced to point Z such that UY ≅

UZ and XS̅̅ ̅ is also produced to point Z. Show that XS ≅ ZS by completing the table below the

triangle.

Statement Reason

TY ∥ SU

U is the midpoint of YZ if follows that S is

also the midpoint of XZ

Hence XS ≅ ZS

X

Y Z

T

U

S



Answer keys Day 67:

Statements Reasons

TY ∥ SU Opposite sides of a parallelogram are parallel

U is the midpoint of YZ if follows that S is

also the midpoint of XZ

Triangle midpoint theorem

Hence XS ≅ ZS



1. Use the figure below to answer the questions that follow. Points S and T are the midpoints of

sides AC and BC respectively. ST= 2.5𝑖𝑛

a) What is the size of ∆𝐴𝐵𝐶?

b) What is the length of side AB?

127°

A B

C

S T



2. Use the figure below to answer the questions that follow. ∆𝐴𝐵𝐶 is dilated to form ∆𝐶𝐷𝐸.

a) Find the scale factor of dilation.

b) Where is the center of dilation?

c) If side AC is 2.8𝑖𝑛 long, what is lethe ngth of CD?

A

B

C

D

E

1.5𝑖𝑛 3𝑖𝑛



Answer Key

Day 68:

1. a) 53°

b) 5𝑖𝑛

2. a) 1

2

b) Point C

c) 1.4𝑖𝑛



1. Plot a graph with a scale of 1square representing 1 unit as shown below

2. Draw a triangle with its vertices at points (0,0) 𝐵(5,1), 𝐶(1,4).

3. Identify the midpoints of sides AB and AC and label them as D and E respectively.

4. Join D and E with a straight line.

What are the vertices of ∆𝐴𝐷𝐸?

5. Dilate ∆𝐴𝐵𝐶 with a scale factor of 1

2.

Does the triangle resulting from the dilation coincide with ∆𝐴𝐷𝐸?

-6 -4 -2 0 2 4 6 x

y

4

2

-2

-4



In this activity, students will draw a triangle and a line passing through midpoints of two sides

and compare the results with a dilation of the same triangle with a scale factor of 0.5. Students

will work in groups of at least three and each group is required to have a graph paper, a pencil

and a ruler.

Answer Keys

Day 68:

1-3. No response

4. 𝐴(0,0), 𝐷(2.5,0.5), 𝐸(0.5,2)

5. Yes



Use the diagram below to answer questions 1-5.

AB is parallel to ST. 𝑆𝑈 = 8𝑖𝑛, ST= 6𝑖𝑛, and AB= 3𝑖𝑛. (the figure is not drawn to scale)

1. Which transformation will map ∆𝑆𝑇𝑈 onto ∆𝐴𝐵𝑈?

2. What is the length of AU?

3. What is the length of AS?

4. What is the length of BU?

5. What is the length of BT?


𝐴𝐶 ∥ 𝑅𝑇 and RT is twice the length of RT.

6. Which transformation will map ∆𝐴𝐵𝐶 onto ∆𝑇𝐵𝑅?

𝑆 T

U

A B

6 𝑖𝑛

3𝑖𝑛

5𝑖𝑛

𝑅

A

B C

T



7. What is the length of CR?

8. What is the length of BR?

9. What is the length of BT?

10. What is the length of AT?

Use the figure below to answer questions 11 to 15.

RQ is parallel to MN and NO=MO=10 in.

11. Which geometric transformation will map ∆𝑀𝑁𝑂 onto ∆𝑄𝑅𝑂?

12. What is the length of RO?

13. What is the length of MR?

14. What is the length of OQ?

15. What is the length of NQ?

12𝑖𝑛

Q R

O

N M

6𝑖𝑛




𝐾𝐽 =1

2𝐵𝐶.

16. Which geometric transformation will map ∆𝐴𝐵𝐶 onto ∆𝐴𝐽𝐾?

17. Write an equation that relates AK and CK.

18. Write an equation that relates AK and AC.

19. Write an equation that relates AJ and BJ.

20. Write an equation that relates BJ and AB.

A B

C

K

J



Answer Keys

Day 68:

1. A dilation about point U with a scale factor of 1

2

2. 4𝑖𝑛

3. 4𝑖𝑛

4. 5𝑖𝑛

5. 5𝑖𝑛

6. A dilation about point B with a scale factor of 1

2

7. 1.5𝑖𝑛

8. 1.5𝑖𝑛

9. 2.5𝑖𝑛

10. 2.5𝑖𝑛

11. A dilation about point B with a scale factor of 1

2

12. 5𝑖𝑛

13. 5𝑖𝑛

14. 5𝑖𝑛

15. 5𝑖𝑛

16. A dilation about point C with a scale factor of 1

2

17. 𝐴𝐾 = 𝐶𝐾

18. 𝐶𝐾 =1

2𝐴𝐶

19. 𝐴𝐽 = 𝐵𝐽

20. 𝐵𝐽 =1

2𝐴𝐵



1. Points D and E are midpoints of AB and BC respectively. ∆𝐵𝐷𝐸 is a dilation of ∆𝐴𝐵𝐶. What

is the scale factor of dilation?

A B

C

D

E



Answer Keys

Day 68:

1. 1

2



Use the following diagram to answer the questions

Line AJ and FH are parallel. Line NG = 3.5 in and NG = 2MG. Angle GKH =116° and Angle

AMS = 42°.

1. Find the length of MN

2. Find the size of angle GTJ.

3. Find the size of angle GTJ.

4. Find the size of angle GNH.

5. Find the length of MN

A

F

G

H

J T

M

N

K

S



Answer Keys

Day 69:

1. 3.5 in

2. 116°

3. 42°

4. 42°

5. 7 in



1. Using 3 rods make a triangle of suitable size by connecting the rods with strings.

2. Label the vertices of the triangle as ABC and measure their length.

3. What are the dimensions of the triangle?

4. Pick any two sides and identify their midpoint using a ruler

5. Tie the last rod to form a line from one midpoint to the other.

6. Measure the distance between these midpoints



7. Compare the distance in 6 above with the third side of the triangle ( the side whose midpoint

was not identified in 4 above).

8. Find the shortest distance between the third side in 7 above and the line connecting the

midpoints.

9. Is the shortest distance constant throughout the two rods?

10. What do you conclude based on 9 above?



In this activity, students will work in groups of at least 4. They will create a framework from

metal, plastic or wooden thin rods and verify the midpoint theorem. Each group will require a 4

five metal, wooden or plastic rods or length 5 – 7 inches each, strings and a ruler.

Answer Keys

Day 69:

1. No response

2. The vertices are labeled

3. Answers vary

4 -5.No response

6. Answer varies

7. The third side must be approximately twice line connecting the midpoint

8. No response

9. Yes

10. The Third side and the line connecting the midpoints are parallel



Use the following information to answer questions 1 – 8

In the figure below, N divides LQ in the ratio of 1:1 while PM = 𝑃𝑀 = 𝑀𝑄. A point T is on line

PL such that 𝑃𝐿 = 2𝑇𝐿. 𝑇𝐿 = 3 𝑖𝑛, 𝐿𝑁 = 5.5 𝑖𝑛 and 𝑃𝑄 = 9 𝑖𝑛. Angle 𝑃𝑇𝑀 = 27°.

1. Which kind of lines are PL and MN?

2. Find the relationship between the two lines above.

3. What is the measurement of MN.

4. What is the measurement of PL.

5. What is the measurement of MQ

6. What is the measurement of LQ.

7. What kind of lines are MN and TL if any.

8. Is there any linear relationship between the two lines in 7 above?

9. What kind of figure is TMNL?

P

Q

L

M

N



10. Explain your results above.

11. Find the size of angle LNM.

12. Find the side of angle MNQ

13. Find the size of angle PLN.

Use the following diagram to answer the following questions. 14 – 20.

In the diagram below, HK and D are midpoints of AG, GC, and CE respectively.

14. Compare the length of HK and AC.

15. Compare the length of HK and AB.

16. Compare the length of KD and GE.

17. Provide the reason for your answer in 16 above.

18. If HKD is a straight line and K divides HD twice, what type of lines are HD and GE?

19. What is the linear relationship between GE and AC.

20. What kind of figure is KDEF.

A B C

D

E F G

H

K



Answer keys Day 69:

1. Parallel lines

2. 𝑃𝐿 = 2𝑀𝑁

3. 3 in

4. 6 in

5. 4.5 in

6. 11 in

7. They are parallel

8. 𝑀𝑁 = 𝑇𝐿

9. Parallelogram

10. MN and TL are parallel while LN and TM are parallel too, Opposite sides are parallel

11. 153°

12. 27°

13. 27°

14. 𝐻𝐾 =1

2𝐴𝐶

15. 𝐻𝐾 = 𝐴𝐵

16. 2𝐾𝐷 = 𝐺𝐸

17. Due to mid-point theorem since K and D are midpoints of CE and CG respectively

18. 𝐻𝐷 = 𝐺𝐸

19. 𝐺𝐸 = 𝐴𝐶

20. Parallelogram



In the figure below, R and E are the midpoints of YP and YH respectively. Find the size of angle

REH.

P H

Y

R E

43°



Answer Keys

Day 69

137°

1. Construct a perpendicular bisector to the given lines. a) · 2020. 3. 3. · Day 66 Practice Name _____ HighSchoolMathTeachers@2020 Page 13 18. Write the equation to relate RK

Documents