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5.1 and 5.4 Perpendicular and Angle Bisectors & ∆ Midsegment Theorem

THEOREMS:

1) If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the

endpoints of the segment.

Write the converse:

2) If a point is equidistant from the sides of an angle of a triangle, then the point lies on the bisector of the

angle.

Write the converse:

The mid-segment of a triangle is a segment joining the __________________of two sides of a triangle.

Properties of a mid segment:

1. is ____________ to the third side

2. is ____________ as long as the third side.

Example 1: Use the points A(2,2) B(12,2), and C(4,8) for the following.

1) Find X and Y, the midpoints of AC and CB.

2) Find XY and AB

3) Find the slope of AB and XY

4) What is the slope of a line parallel to 3x + 2y = 12?

M, N , and P are midpoints of ,,ZYXZ and XY , respectively.

1.) Mark the diagram with tick marks:

2) Name all ≅ ∆’s:

3) XY // _____; XZ // _____; MP // ____

Example 1) Given DE, DF, and FE are the lengths of Example 2) Given AC = 42, CB = 46,

mid-segments. Find the perimeter of triangle ABC. AB = 48, D, E, and F are midpoints Find the

perimeter of triangle DEF

Example 3) D and E are midpoints. Find m<A and Example 4) Find the value of x. The diagram

m< EDA. Is NOT drawn to scale.

Z

M N

X Y P

2

Example 5) Find the value of x. Example 6) Points B, D, and F are midpoints.

EC = 30 and DF = 23. Find AC.

Example 7) Identify the mid-segment and Example 8) If BE = 2x+6 and DF = 5x+9,

find its length. find the value of x, DF, and BE

Example 9) Q is equidistant from the sides of Find

the value of x. The diagram is not to scale.

Use the figure for Exercises 2–5.

2. Given that line p is the perpendicular bisector of

XZ and XY 15.5, find ZY. _____________________

3. Given that XZ 38, YX 27, and YZ 27,

find ZW. _____________________

4. Given that line p is the perpendicular bisector of ;XZ XY 4n,

and YZ 14, find n. _____________________

5. Given that XY ZY, WX 6x – 1, and XZ 10x 16, find ZW. _____________________

Use the figure for Exercises 6–9.

6. Given that FG HG and mFEH 558, find

mGEH. _____________________

7. Given that EG bisects FEH and 2,GF find GH.

_____________________

8. Given that FEG GEH, FG 10z – 30, and

HG 7z 6, find FG. _____________________

9. Given that GF GH, mGEF 8

3 a8, and mGEH 248, find a. ____________________

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Many plants grow in geometric patterns. The figure shows the veins in a leaf

from an alder tree. Refer to the figure for Exercises 2–5. Match the letter of each

theorem to the statement that uses the theorem.

________ 2. If BD CD, then D is on

the bisector of BAC.

________ 3. If BAD CAD, then

BD CD.

________ 4. If QP RP and ,SP QR

then QS RS.

________ 5. If QS RS and QP RP,

then .SP QR

Use the figure for Exercises 6 and 7.

6. Given that line m is the perpendicular bisector of

FH and EH 100, find EF. _________________

7. Given that EF 13, FH 10, and EH 13, find GH. _________________

Use the figure for Exercises 8 and 9.

8. Given that JL bisects KJM and KL 42, find ML. _________________

9. Given that KL 4 and ML 4 and mMJL 408, find

mKJL. _________________

A. Perpendicular Bisector Theorem

B. Converse of the Perpendicular Bisector Theorem

C. Angle Bisector Theorem

D. Converse of the Angle Bisector Theorem

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5.2 and 5.3 Special Segments of a :

Def: A median of a triangle is __________________________________________________

Draw the three medians of the The point of concurrency of the three

following triangle: medians of a triangle is

called the _______________.

Def : An altitude of a triangle is ___________________________________________________

Draw the altitude from all three vertices:

The point of concurrency of the lines containing the altitudes of a triangle is called the

__________________.

Def An angle bisector is ____________________________________________________

Draw the angle bisectors of all 3 vertices:

The point of concurrency of the < bisectors of the angles of a triangle is called the ________________.

Def Perpendicular bisector of a segment:________________________________________.

Draw the perpendicular bisectors of each side of the :

The point of concurrency of the bisectors of the sides of a triangle is called the __________________

***Note: Angle bisectors, medians and altitudes always have a vertex as one endpoint. This is not

always the case for perpendicular bisectors.

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Centroid Theorem– The medians of a ∆ are concurrent at a point that is 2/3

the distance from each vertex to the midpoint of the opposite side.

Example 1) D is the centroid of ∆ABC and DE = 6. Find BD and BE.

If AD = 9, find DF and AF.

Example 2) name the median, altitude, and angle bisector

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Practice with Altitudes and Medians Each figure below shows one or more medians.

1.

Find x. DB is a median. SU is a median. Find x.

In PRS, 𝑷𝑻̅̅ ̅̅ is an altitude and 𝑷𝑿̅̅ ̅̅ is a median.

2. Find RS if RX = x + 7 and SX = 3x – 11.

3. Find RT if RT = x – 6 and mPTR = 8x – 6.

4. Find x if 𝐸𝐺̅̅ ̅̅ is a median of DEF.

Find the coordinates of the centroid of each triangle.

5.

8

Find the value of each variable.

1) 2)

3) 4)

5)

Find the coordinates of the centroid of each triangle given the three vertices.

6) 7)

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5.5 : and 5.6 Inequalities (1 and 2 Triangles)

In a , the smallest is opposite the shortest side.

In a , the largest is opposite the longest side.

Converses are also true!

In a , the shortest side is opposite the smallest .

In a , the longest side is opposite the largest .

1. In ∆ ABC name the sides in order from least to greatest.

2. In ∆ DEF name the angles in order from greatest to least.

3. Name the shortest and longest sides in right ∆ FIT if F is the right angle and mI = 48.

Theorem The sum of the lengths of any 2 sides of a triangle is greater than the length of the 3rd side.

8. Can a triangle have sides with the given lengths?

a) 4m, 7m, 8m d) 1.2cm, 2.6cm, 4.9cm

b) 4in, 4in, 4in e) 11m, 12m, 14m

c) 18ft, 20ft, 40ft f) 2.5m, 3.5m, 6m

TRIANGLE INEQUALITIES

II. In the following exercises the diagrams are not drawn to scale. If each diagram were drawn

to scale, which numbered angle would be largest?

1. 2. 3. Which segment would be

the largest?

The lengths of two sides of a triangle are given. Find the range of possible lengths

for the third side.

10. 8.2 m, 3.5 m 11. 298 ft, 177 ft 12. 1

32

mi, 4 mi

B C

D

E

F

54

52

12mm

13mm

11mm

13 12

12

1 2

3

x

x+ 1

x - 1 1

3

2

10

The diagrams are not drawn to scale. Which numbered angle would be the largest?

5. 6.

_____________ ______________

Which segment is the longest?

7. 8.

__________________ __________________

Use lengths to complete:

9.

_________ > _________ > _________

6 cm 5 cm

4 cm

1 2

3 y + 2 y

y - 2

2 1

3

63

58

C

B A

c

61

59

b a

(x + 1) (x + 3)

60

C A

B

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Compare the given measures.

1. mK and mM 2. AB and DE 3. QR and ST

Find the range of values for x.

4.

6.

5.

7.