Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency ...

Triangles: Points of Concurrency

MM1G3 e

Investigate Points of Concurrency

• http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html

Circumcenter

Perpendicular Bisectors and Circumcenters Examples

A perpendicular bisector of a triangle is a line or line segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisector of the side.

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Example 1:

Example 2:

Q

P N

M

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In ΔMNP above, MQ ≅ NQ and PQ⊥MN .

Therefore, PQ is the perpendicular bisector of MN .

*Notice that PQ contains the vertex P.

Since a triangle has three sides, it will have three perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter.

FE

G

D

G is the circumcenter of ∆DEF.

Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices.

. So, FGEGDG

The circumcenter will be located inside an acute triangle (fig.1), outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters.

fig. 1 fig. 3

fig. 2

Example 3: A company plans to build a new distribution center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?

The circumcenter is equidistant to all three vertices of a triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same.

C

In Summary

• The circumcenter is the point where the three perpendicular bisectors of a triangle intersect.

• The circumcenter can be inside, outside, or on the triangle.

• The circumcenter is equidistant from the vertices of the triangle

Circumcenter

• Exploration

• Construction

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Medians and Centroids Examples

A median of a triangle is a line segment that contains the vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.

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and ofmidpoint theis Therefore, . bisects above, In

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Since a triangle has three sides, it will have three medians. These medians will meet at a common point – the centroid.

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The centroid is always located inside the triangle.

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The distance from any vertex to the centroid is 2/3 the length of the median. Q

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of 32 is centroid the to vertex from

distance The median.a is , In

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Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3. Find QD and SF.

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QD

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. then, Since

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Example 3: G is the centroid of triangle DEF. FG = 15, ES = 21, QG = 5 Determine FR, EG and GD

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R

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F G

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EG

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and Since

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GD

GD

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QDGD

Notice that the distance from any vertex to the centroid is 2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median.

So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ

Q

P

N

U

M

T

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The centroid is also known as the balancing point (center of gravity) of a triangle.

In Summary• A median is a line segment from the a vertex of a triangle

to the midpoint of the opposite side.

• The distance from the vertex to the centroid is 2/3 the length of the median.

• The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid.

• Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.

Centroids

• Investigate

• Construction

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Angle Bisectors and Incenters Examples

An angle bisector of a triangle is a segment that shares a common endpoint with an angle and divides the angle into two equal parts.

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Example 1: Determine any angle bisectors of triangle ABC.

A

BC

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E

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bisector angle an is Therefore,

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Since a triangle has three angles, it will have three angle bisectors. These angle bisectors will meet at a common point – the incenter.

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X

triangle. theofincenter theis Therefore,

.point at intersect bisectors angle three the, In

M

MXYZ

The incenter is always located inside the triangle.

Acute triangle

Right triangle

Obtuse triangle

incenter

The incenter is equidistant to the sides of the triangle.

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and ,,, In

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Example 2: L is the incenter of triangle ABC. Which segments are congruent?

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B

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Example 3: Given P is the incenter of triangle RST. PN = 10 and MT = 12, find PM and PT.

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In Summary

• The incenter is the point of intersection of the three angle bisectors of a triangle.

• The incenter is equidistant to all three sides of the triangle.

Incenter

• Investigate

• Construction

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Altitudes and Orthocenters Examples

An altitude of a triangle is the perpendicular distance between a vertex and the opposite side. This distance is also known as the height of the triangle.

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and lar toperpendicu is that Notice . of altitude an is

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Example 1: Determine any altitudes of triangle ABC.

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D

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. vertex, thecontains and lar toperpendicu is . vertex, thecontains

and lar toperpendicu is altitudes. are and only , In

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BCAFCDAFABC

Since a triangle has three sides, it will have three altitudes. These altitudes will meet at a common point – the orthocenter.

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X

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. point, common at themeet altitudes These

triangle. theof altitude an each are and ,, that Notice

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The orthocenter may be located inside an acute triangle (fig. 1), outside an obtuse triangle (fig. 2), or on a right triangle (fig. 3). In the triangles below, the red lines represent altitudes. The red points indicate the orthocenters.

Obtuse TriangleAcute Triangle Right Triangle

Fig. 3Fig. 2Fig. 1

Summary• An altitude is a line segment containing a

vertex of a triangle and is perpendicular to the opposite side.

• The orthocenter is the intersection point of the three altitudes of a triangle.

• Orthocenters can be inside, outside, or on the triangle depending on the type of triangle.

Orthocenter

• Investigate

• Construction

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Points of Concurrency

Investigate

Points of Concurrency

MM1G3 eReview

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