- 161 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Unit 6' - amable

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Unit 6

Properties and Attributes of Triangles

5.1 Perpendicular and Angle Bisectors

Essential Question

Compare and Contrast Perpendicular Bisectors and Angle bisectors.

Example 1A: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure.

MN

MN = LN

Bisector Thm.

MN = 2.6

Substitute 2.6 for LN.

Example 1B: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure.

BC

Since AB = AC and , is the perpendicular bisector of by the Converse of the Perpendicular Bisector Theorem.

BC = 2CD

Definition of Perpendicular bisector.

BC = 2(12) = 24

Substitute 12 for CD.

Example 1C: Applying the Perpendicular Bisector Theorem and Its Converse

Find each measure.

TU

TU = UV

Bisector Thm.

3x + 9 = 7x – 17

Substitute the given values.

9 = 4x – 17

Subtract 3x from both sides.

26 = 4x

Add 17 to both sides.

6.5 = x

Divide both sides by 4.

So TU = 3(6.5) + 9 = 28.5.

Helpful Hint Its Converse

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

HINTPerpendicular Bisectors intersect at the Circumcenter Its Converse

When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.

The Its Conversecircumcenter can be inside the triangle, outside the triangle, or on the triangle.

The Its Conversecircumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

DG Its Converse, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.

Example 1: Using Properties of Perpendicular BisectorsG is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of

∆ABC.

GC = CB

Circumcenter Thm.

Substitute 13.4 for GB.

GC = 13.4

MZ Its Converse is a perpendicular bisector of ∆GHJ.

Check It Out! Example 1aUse the diagram. Find GM.

GM = MJ

Circumcenter Thm.

Substitute 14.5 for MJ.

GM = 14.5

KZ Its Converse is a perpendicular bisector of ∆GHJ.

Check It Out! Example 1bUse the diagram. Find GK.

GK = KH

Circumcenter Thm.

Substitute 18.6 for KH.

GK = 18.6

Check It Out! Its Converse Example 1c

Use the diagram. Find JZ.

Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of

∆GHJ.

JZ = GZ

Circumcenter Thm.

Substitute 19.9 for GZ.

JZ = 19.9

Angle Bisectors Its Converse

- Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.

Example 2A: Applying the Angle Bisector the locus of all points in the interior of the angle that are equidistant from the sides of the angle.Theorem

Find the measure.

BC

BC = DC

Bisector Thm.

BC = 7.2

Substitute 7.2 for DC.

Since the locus of all points in the interior of the angle that are equidistant from the sides of the angle.EH = GH,

and , bisects

EFGby the Converse

of the Angle Bisector Theorem.

Example 2B: Applying the Angle Bisector TheoremFind the measure.

mEFH, given that mEFG = 50°.

Def. of bisector

Substitute 50° for mEFG.

Since, the locus of all points in the interior of the angle that are equidistant from the sides of the angle.JM = LM, and

, bisects JKL

by the Converse of the Angle

Bisector Theorem.

Example 2C: Applying the Angle Bisector TheoremFind mMKL.

mMKL = mJKM

Def. of bisector

3a + 20 = 2a + 26

Substitute the given values.

a + 20 = 26

Subtract 2a from both sides.

a= 6

Subtract 20 from both sides.

So mMKL = [2(6) + 26]° = 38°

A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle.

Remember! bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

The distance between a point and a line is the length of the perpendicular segment from the point to the line.

Remember!Unlike the bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the circumcenter, the incenter is always inside the triangle.

The incenter is the center of the triangle’s bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the inscribed circle. A circle inscribedin a polygon intersects each line that contains a side of the polygon at exactly one point.

MP bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the and LP are angle bisectors of ∆LMN. Find the distance from P to MN.

The distance from P to LM is 5. So the distance from P to MN is also 5.

Example 3A: Using Properties of Angle BisectorsP is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN.

PL is the bisector of bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the MLN.

PM is the bisector of LMN.

Example 3B: Using Properties of Angle BisectorsMP and LP are angle bisectors of ∆LMN. Find mPMN.

mMLN = 2mPLN

mMLN = 2(50°)= 100°

Substitute 50° for mPLN.

mMLN + mLNM + mLMN = 180°

ΔSum Thm.

100+ 20 + mLMN = 180

Substitute the given values.

Subtract 120° from both sides.

mLMN= 60°

Substitute 60° for mLMN.

5.1 Perpendicular and Angle Bisectors bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

Summary

Compare and Contrast Perpendicular Bisectors and Angle bisectors.

The perpendicular bisector____________.

Perpendicular bisectors intersect at ________ point of concurrency.

The angle bisector____________.

Angle bisectors intersect at ________ point of concurrency.

Both the perpendicular bisector and angle bisector ____________________________.

Median bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the and Altitude of a Triangle

How can I find the centroid and orthocenter?

Medians intersect at the Centroid bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

Altitudes Intersect at the Orthocenter bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

Median, Altitude and Midsegment bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

I can find the centroid by____________.

I can find the orthocenter by _______________.

Midsegment of a Triangle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

How does 2/3 relate to Midsegment of a triangle?

Midsegment bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

Triangle Midsegment Theorem bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

Midsegment of a Triangle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the

2/3 of a ___________ represents the ______ of a Triangle.

Download Presentation

Connecting to Server..