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Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the...

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Geometry Unit 5: Triangle Parts
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Page 1: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

GeometryUnit 5: Triangle Parts

Page 2: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

CONCURRENT:

Page 3: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Concurrent:

When three or more lines intersect at the same point, P

P

Page 4: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MIDSEGMENT:

Page 5: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Triangle Midsegment:

Segment connecting the midpoints of two sides of a triangle

Page 6: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Triangle Midsegments

• Parallel to the third side of the triangle

Page 7: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Triangle Midsegments

• Parallel to the third side of the triangle

• Half the length of the third side of the triangle

Page 8: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

BISECTORS

Page 9: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Bisectors

Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.

Page 10: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Bisectors

Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.

• The circles will be inscribed in or circumscribed about triangles.

Page 11: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTORS

Page 12: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Lines that bisect a side and are perpendicular to it

Page 13: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTOR

Bisects a side and makes a 90 angle with it

Page 14: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors (in purple)

• Concurrent at the CIRCUMCENTER of each green triangle

Page 15: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Circumcenter can be inside, on, or outside of the triangle

Page 16: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Circle 1: circumcenter is outside triangle• Circle 2: circumcenter is inside triangle

. .circumcenter

circumcenter

Page 17: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• purple radii of circle go from circumcenter to each vertex of the triangle

r

r

Page 18: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Can you identify three isosceles triangles in each figure?

r

r

Page 19: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Lines that bisect a side and are perpendicular to it

• Concurrent at the Circumcenter of the triangle

• Circumcenter can be inside, on, or outside of the triangle

• Radii of circle go from circumcenter to each vertex of the triangle

Page 20: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle Bisector

Page 21: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle Bisector

Bisects the angle

Page 22: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle BisectorConcurrent at the INCENTER

(center of circle INSCRIBED in triangle)

Page 23: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle BisectorConcurrent at the INCENTER

(center of circle INSCRIBED in triangle)

Page 24: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

Page 25: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIANSegment from vertex to opposite

side’s midpoint

(Nothing to do with the angle!!)

Page 26: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

Concurrent at CENTROID

(center of gravity)

Page 27: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

The center of gravity is the BALANCE POINT.

Page 28: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

The CENTROID must be inside of the triangle!

Page 29: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDE

Page 30: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDE

Perpendicular segment from a vertex to the side opposite (or extension of the side

opposite).

Page 31: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDE

Height of the triangle

(perpendicular to the base)

Page 32: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDES

Page 33: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDES

Concurrent at the ORTHOCENTER

orthocenter.

Page 34: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDES

Can be outside of a triangle.

altitude

Page 35: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Quick Review

Page 36: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

Page 37: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

MEDIAN

Concurrent at CENTROID

(center of gravity)

. Centroid

Page 38: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle Bisector

Page 39: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle BisectorConcurrent at the INCENTER

(center of circle INSCRIBED in triangle)

Page 40: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Angle BisectorConcurrent at the INCENTER

(center of circle INSCRIBED in triangle)

. Incenter

Page 41: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDE

Page 42: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

ALTITUDES

Concurrent at the ORTHOCENTER

orthocenter.

Page 43: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTORS

Page 44: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors:

Concurrent at the Circumcenter

. circumcenter

Page 45: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Perpendicular Bisectors

• Radii of circle go from circumcenter to each vertex of the triangle

r

r

r

Page 46: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

BISECTORS

Page 47: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Bisectors

Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.

Page 48: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Problem: Three cities want to build a park that is the same distance from each of their city centers. What should they do?

Kenmore

Shoreline

MLT

Page 49: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which “triangle center” will be the same distance from each city center?

Shoreline

Kenmore

Shoreline

MLT

Page 50: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The CIRCUMCENTER

Shoreline

Kenmore

Shoreline

MLT

Page 51: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which triangle segments or lines are used to find the circumcenter?

Shoreline

Kenmore

Shoreline

MLT

Page 52: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTORS

Shoreline

Kenmore

Shoreline

MLT

Page 53: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTORS(green lines) are concurrent at the

CIRCUMCENTER.

Kenmore

MLT

CircumcenterShoreline

Page 54: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The Circumcenter is equidistant from each city center.

Kenmore

MLT

Circumcenter

Shoreline

Page 55: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The distance is the RADIUS of the circle centered at the CIRCUMCENTER.

Kenmore

MLT

C

Shoreline

r

r r

Page 56: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Problem: Three cities want to build a toxic waste dump that is the same distance from each of their city centers. What should they do?

MLTKenmore

Shoreline

Page 57: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which “triangle center” will be the same distance from each city center?

MLTKenmore

Shoreline

Page 58: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which “triangle center” will be the same distance from each city center? The CIRCUMCENTER

MLTKenmore

Shoreline

Page 59: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which triangle segments or lines are used to find the circumcenter?

MLTKenmore

Shoreline

Page 60: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

Which triangle segments or lines are used to find the circumcenter?

PERPENDICULAR BISECTORS

MLTKenmore

Shoreline

Page 61: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

PERPENDICULAR BISECTORSare concurrent at the CIRCUMCENTER.

Kenmore

Shoreline

MLT

Circumcenter

Page 62: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The Circumcenter is equidistant from each city center.

KenmoreShoreline

MLT

Circumcenter

Page 63: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The distance is the RADIUS of the circle centered at the CIRCUMCENTER.

KenmoreShoreline

MLT

Circumcenter

radius

Page 64: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The circumcenter can be outside of the triangle.

KenmoreShoreline

MLT

Circumcenter

radius

Page 65: Geometry Unit 5: Triangle Parts. CONCURRENT: Concurrent: When three or more lines intersect at the same point, P P.

The centroid and incenter must be inside

of the triangle.


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