Vocabulary MedianA segment with endpoints being a vertex of a
triangle and the midpoint of the opposite side. AltitudeA segment
from a vertex to the line containing the opposite side and
perpendicular to the line containing that side. An altitude may be
out side the circle.
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Concept (also called center of mass or center of gravity) From
Latin: centrum - "center", and Greek: -oid -"like"
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Example 1 Use the Centroid Theorem In XYZ, P is the centroid
and YV = 12. Find YP and PV. Centroid Theorem YV = 12
Simplify.
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Example 1 Use the Centroid Theorem Answer: YP = 8; PV = 4 YP +
PV= YVSegment Addition 8 + PV= 12YP = 8 PV= 4Subtract 8 from each
side.
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A.A B.B C.C D.D Example 1 A.LR = 15; RO = 15 B.LR = 20; RO = 10
C.LR = 17; RO = 13 D.LR = 18; RO = 12 In LNP, R is the centroid and
LO = 30. Find LR and RO.
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Example 2 Use the Centroid Theorem In ABC, CG = 4. Find
GE.
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Example 2 Use the Centroid Theorem Centroid Theorem Segment
Addition and Substitution CG = 4 Distributive Property
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Example 2 Use the Centroid Theorem Answer: GE = 2 Subtract GE
from each side. __ 1 3
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A.A B.B C.C D.D Example 2 A.4 B.6 C.16 D.8 In JLN, JP = 16.
Find PM.
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Example 3 Find the Centroid on a Coordinate Plane SCULPTURE An
artist is designing a sculpture that balances a triangle on top of
a pole. In the artists design on the coordinate plane, the vertices
are located at (1, 4), (3, 0), and (3, 8). What are the coordinates
of the point where the artist should place the pole under the
triangle so that it will balance? UnderstandYou need to find the
centroid of the triangle. This is the point at which the triangle
will balance.
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Example 3 Find the Centroid on a Coordinate Plane SolveGraph
ABC. PlanGraph and label the triangle with vertices (1, 4), (3, 0),
and (3, 8). Use the Midpoint Theorem to find the midpoint of one of
the sides of the triangle. The centroid is two-thirds the distance
from the opposite vertex to that midpoint.
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Example 3 Find the Centroid on a Coordinate Plane Find the
midpoint D of side BC. Graph point D.
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Example 3 Find the Centroid on a Coordinate Plane Notice that
is a horizontal line. The distance from D(3, 4 ) to A(1, 4) is 3 1
or 2 units.
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Example 3 Find the Centroid on a Coordinate Plane The centroid
is the distance. So, the centroid is (2) or units to the right of
A. The coordinates are.
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Example 3 Find the Centroid on a Coordinate Plane Answer: The
artist should place the pole at the point CheckCheck the distance
of the centroid from point D (3, 4). The centroid should be (2) or
units to the left of D. So, the coordinates of the centroid is. __
1 3 2 3
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A.A B.B C.C D.D Example 3 BASEBALL A fan of a local baseball
team is designing a triangular sign for the upcoming game. In his
design on the coordinate plane, the vertices are located at (3, 2),
(1, 2), and (1, 6). What are the coordinates of the point where the
fan should place the pole under the triangle so that it will
balance? A.(, 2) B.(, 2) C.(1, 2) D.(0, 4) __ 7 3 5 3
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Concept From Greek: orthos - "straight, true, correct,
regular
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Example 4 Find the Orthocenter on a Coordinate Plane COORDINATE
GEOMETRY The vertices of HIJ are H(1, 2), I(3, 3), and J(5, 1).
Find the coordinates of the orthocenter of HIJ.
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Example 4 Find the Orthocenter on a Coordinate Plane Find an
equation of the altitude from The slope of so the slope of an
altitude is Point-slope form Distributive Property Add 1 to each
side.
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Example 4 Find the Orthocenter on a Coordinate Plane
Point-slope form Distributive Property Subtract 3 from each side.
Next, find an equation of the altitude from I to The slope of so
the slope of an altitude is 6.
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Example 4 Find the Orthocenter on a Coordinate Plane Equation
of altitude from J Multiply each side by 5. Add 105 to each side.
Add 4x to each side. Divide each side by 26. Substitution, Then,
solve a system of equations to find the point of intersection of
the altitudes.
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Example 4 Find the Orthocenter on a Coordinate Plane Replace x
with in one of the equations to find the y-coordinate. Multiply and
simplify. Rename as improper fractions.
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Example 4 Find the Orthocenter on a Coordinate Plane Answer:
The coordinates of the orthocenter of HIJ are
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A.A B.B C.C D.D Example 4 A.(1, 0) B.(0, 1) C.(1, 1) D.(0, 0)
COORDINATE GEOMETRY The vertices of ABC are A(2, 2), B(4, 4), and
C(1, 2). Find the coordinates of the orthocenter of ABC.