Jim Smith JCHSJim Smith JCHS

UNIT 5.4 MEDIANS UNIT 5.4 MEDIANS AND ALTITUDESAND ALTITUDES

There are 3 of each of these specialThere are 3 of each of these special segments in a triangle.segments in a triangle.

The 3 segments are concurrent. TheyThe 3 segments are concurrent. Theyintersect at the same point.intersect at the same point.

This point is called the point of This point is called the point of concurrency.concurrency.

The points have special names and The points have special names and special properties.special properties.

Start at a vertex and form a Start at a vertex and form a 9090° angle° anglewith the line containing the opposite side.with the line containing the opposite side.

OrthocenterOrthocenter

The The orthocenterorthocenter can be located can be locatedin the triangle, on the triangle orin the triangle, on the triangle oroutside the triangle.outside the triangle.

RightRight

Legs are altitudesLegs are altitudes

ObtuseObtuse

Angle bisectors start at a vertexAngle bisectors start at a vertexand and bisect the angle.bisect the angle.

IncenterIncenter

Any point on an angle bisector is Any point on an angle bisector is equidistance from the sides of the angle equidistance from the sides of the angle

This makes the This makes the IncenterIncenter an anequidistance from all 3 sidesequidistance from all 3 sides

A Perpendicular bisector of a side does A Perpendicular bisector of a side does not have to start at a vertex. It will formnot have to start at a vertex. It will form

a a 9090° angles° angles and bisectand bisect the side. the side.

CircumcenterCircumcenter

Any point on the Any point on the perpendicular bisectorperpendicular bisectorof a segment is equidistance from theof a segment is equidistance from the

endpoints of the segment.endpoints of the segment.

AA

BB

CC DD

AB is the AB is the perpendicularperpendicularbisector of CDbisector of CD

This makes the This makes the CircumcenterCircumcenter an anequidistance from the 3 verticesequidistance from the 3 vertices

The median starts at a vertex and endsThe median starts at a vertex and endsat the at the midpointmidpoint of the opposite side. of the opposite side.

CentroidCentroid

The The centroidcentroid is 2/3’s of the distance is 2/3’s of the distance from the vertex to the side.from the vertex to the side.

2x2x

xx

3232

1616

The The centroidcentroid is the center of balance is the center of balance for the triangle. You canfor the triangle. You can

balance a triangle on the tip ofbalance a triangle on the tip ofyour pencil if you place the tip onyour pencil if you place the tip on

the the centroidcentroid

Altitude ..Altitude .. Vertex .. 90Vertex .. 90°° .. .. OrthocenterOrthocenter

Angle BisectorAngle Bisector.... Angle into 2 equal angles .. Angle into 2 equal angles .. IncenterIncenter

Perpendicular BisectorPerpendicular Bisector…… 90° .. bisects side .. 90° .. bisects side .. CircumcenterCircumcenter

MedianMedian .. .. Vertex .. Midpoint of side ..Vertex .. Midpoint of side ..CentroidCentroid

Give the best name for ABGive the best name for ABAA

BB

AA

BB

AA

BB

AA

BB

AA

BB||||

|| ||

||||

MedianMedian AltitudeAltitude None None Angle Angle PerpPerp Bisector Bisector BisectorBisector

Survival TrainingSurvival TrainingYou’re Stranded On A Triangular You’re Stranded On A Triangular

Shaped Island. The Rescue Ship CanShaped Island. The Rescue Ship CanOnly Dock On One Side Of The Island Only Dock On One Side Of The Island

But You Don’t Know Which Side. At But You Don’t Know Which Side. At Which Point Of Concurrency Would Which Point Of Concurrency Would

You Set Up Camp So You Are An You Set Up Camp So You Are An EqualEqual Distance From All 3 Sides?Distance From All 3 Sides?

INCENTERINCENTER

What If The Ship Could OnlyWhat If The Ship Could OnlyDock At One Of The Vertices? Dock At One Of The Vertices? Would You Change The Would You Change The Location Of Your Camp ?Location Of Your Camp ?If So, Where?If So, Where?

YESYES CIRCUMCENTERCIRCUMCENTER

Where would you place a fire hydrant toWhere would you place a fire hydrant tomake it equidistance to the houses andmake it equidistance to the houses and

equidistance to the streets?equidistance to the streets?

ELM

ELM

POSTPOST

ELM

ELM

POSTPOST

Angle bisector for the streetsAngle bisector for the streetsPerpendicular bisector for housesPerpendicular bisector for houses

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