G.5 Triangle Midsegments and Angle Bisectors

5.1 Midsegment Theorem

Multiple Select

Which is true about any midsegment in a triangle?

(Select all that apply.) #2

• The midsegment is parallel to its corresponding side.

• The midsegment is perpendicular to its corresponding side.

• The midsegment is half as long as its corresponding side.

• The midsegment connects the midpoints of two sides of a triangle.

• The midsegment is twice as long as its corresponding side.

Multiple Select

Which statement is always true given BC is the midsegment of triangle ADE? Select all that apply. #25

• $$2AB=AD$$  

• $$\overline{AD}\perp\ \overline{DE}$$  

• $$\overline{AC}=\overline{CE}$$  

• $$\overline{BC}\parallel\overline{DE}$$  

Multiple Choice

Is DE parallel to BC?

• yes

• no

Multiple Choice

Is DE half the length of BC?

• yes

• no

Open Ended

Complete the statement.

$$\overline{JK}$$   $$\parallel$$  _______

Use the Midsegment Theorem to Find Lengths

Construction: Triangles are used for strength in roof trusses. In the diagram, UV and VW are midsegments of Triangle RST. Find UV and RS. Solution on next slide.

Find UV & RS.

UV = 1/2 * RT = 1/2 (90 in. ) = 45 in

RS = 2 * VW = 2(57 in.) = 114 in.

Multiple Choice

What is the length of AB?

• 4

• 8

• 16

• 24

Multiple Choice

Find the length of the segment.

• 5

• 5/2

• 10

Multiple Choice

What is the value of x?

• 10

• 5

• 2

• no solution

Multiple Choice

Find v.

• 33

• 66

• 2v + 66

• v + 33

Multiple Choice

Find T.

• 35

• 5

• 40

• 4.5

Reminder!!!

DE = 1/2(AB)

CE = BE

AD = DC

Multiple Choice

If CE = 12, what is the value of CB?

• 12

• 6

• 24

Multiple Choice

What is the value of y?

• 3

• -3

• 5

• no solutoin

Multiple Choice

WINDOWS: A large triangular window is segmented as shown. In the diagram, DF and EF are midsegments of triangle ABC. Find DF.

• 45

• 90

• 180

Multiple Choice

WINDOWS: A large triangular window is segmented as shown. In the diagram, DF and EF are midsegments of triangle ABC. Find AB.

• 45

• 90

• 180

Multiple Choice

Suppose UW = 81, find ST.

• 81

• 40.5

• 162

Math Spoken Here!

Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

Equidistant: A point is equidistant from two figures if it is the same distance from each figure.

Note that the red line is equidistant from the endpoints of c and is perpendicular to side c.

Need to know...

In a plane:

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

If a point is equidistant from the endpoints of a segment then it is on the perpendicular bisector of the segment.

Take Notice:

Did you notice that when the perpendicular bisector passes through the vertex point and the base, then the legs are congruent?

In an isosceles triangle a perpendicular bisector that passes through the base will always pass through the vertex point leaving two congruent legs.

Look closely at the image.

Multiple Choice

Find HG

• 28

• 14

• 41

• 7

Multiple Choice

• JM = 5

• JM = 10

• JM = 12

• JM = 24

Multiple Select

Select every true statement below.

• BC = 10 ft.

• AD = 36 ft.

• BE = 10 ft.

• DE = 36 ft.

Take note:

The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex.

Math Spoken Here!

Concurrent: When three or more lines rays or segments intersect in the same point.

Point of Concurrency: The point of intersection of the lines, rays, or segments.

An example of the use of concurrency of perpendicular bisectors.

Math Spoken Here!

Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle.

Circumcenter:

The circumcenter (call it P) is equidistant from the three vertices, so P is the center of a circle that passes through all three vertices.

As shown, the location of P depends on the type of triangle. The circle with center P is said to be circumscribed about the triangle.

Multiple Choice

Point P is the circumcenter, which segment is equal to PB

• DA

• AC

• PC

• PF

Multiple Choice

Name the point of concurrency shown.

• Circumcenter

• Incenter

• Supercenter

• Neither

Multiple Choice

What does the word BISECT mean?

• To cut something into more than two pieces

• It is an object on the plane with has two equal sets of parts.

• To cut something into two congruent pieces or in half.

• A shape that is created which has three equal sides. 

Multiple Choice

A point that is the same distance from two figures is ___________________ from the two figures.

• Congruent

• Corresponding

• Vertical

• Equidistant

​Perpendicular Bisector Theorem Proof

You must watch the following video:

https://www.youtube.com/watch?v=ttZvFhEq6cg​

Multiple Choice

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

• True

• False

Multiple Choice

Find the length of AD

• 2

• 4

• 6

• 8

Multiple Choice

Find the value of x.

• 2

• 6

• 8

• 4

Multiple Choice

Find the value of x.

• 8

• 16

• 4

• 32

Multiple Choice

What is UT?

• A line segment

• 3

• 15

Multiple Choice

 In the isosceles triangle $$QRS$$  ,  $$\overline{RT}$$  is the angle bisector of  $$\angle QRS$$ . You want to prove the base angles of this triangle are congruent. You start by stating  $$\angle1\cong\angle2$$  . What is a valid reason for this statement? 

• Reflexive property of congruence

• Definition of angle bisector

• Symmetric property of congruence

• Corresponding angles are congruent

Multiple Choice

Given $$AM=BM$$   and $$AC=BC$$  , what is $$m\angle MBC$$  ?

• 26

• 90

• 64

• 13

Multiple Choice

• Perpendicular Bisector

• Angle Bisector

• Median

• Midsegment

Multiple Choice

What goes from the vertex, bisects the angle at that vertex, and connects to the opposite side?

• Angle Bisector

• Perpendicular Bisector

• Median

• Midsegment