Introduction Think about all the properties of triangles we have learned so far and all the...

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IntroductionThink about all the properties of triangles we have learned so far and all the constructions we are able to perform. What properties exist when the perpendicular bisectors of triangles are constructed? Is there anything special about where the angle bisectors of a triangle intersect? We know triangles have three altitudes, but can determining each one serve any other purpose? How can the midpoints of each side of a triangle help find the center of gravity of a triangle? Each of these questions will be answered as we explore the centers of triangles.

1.9.4: Proving Centers of Triangles

Key Concepts• Every triangle has four centers. • Each center is determined by a different point of

concurrency—the point at which three or more lines intersect.

• These centers are the circumcenter, the incenter, the orthocenter, and the centroid.

Key Concepts, continuedCircumcenters

• The perpendicular bisector is the line that is constructed through the midpoint of a segment. In the case of a triangle, the perpendicular bisectors are the midpoints of each of the sides.

• The three perpendicular bisectors of a triangle are concurrent, or intersect at one point.

Key Concepts, continued• This point of concurrency is called the circumcenter

of the triangle.

• The circumcenter of a triangle is equidistant, or the same distance, from the vertices of the triangle. This is known as the Circumcenter Theorem.

Key Concepts, continued

Theorem

Circumcenter TheoremThe circumcenter of a triangle is equidistant from the vertices of a triangle.

The circumcenter of this triangle is at X.

Key Concepts, continued• The circumcenter can be inside the triangle, outside

the triangle, or even on the triangle depending on the type of triangle.

• The circumcenter is inside acute triangles, outside obtuse triangles, and on the midpoint of the hypotenuse of right triangles.

Key Concepts, continued• Look at the placement of the circumcenter, point X, in

the following examples.

Acute triangle Obtuse triangle Right triangle

X is inside the triangle.

X is outside the triangle.

X is on the midpoint of the hypotenuse.

Key Concepts, continued• The circumcenter of a triangle is also the center of the

circle that connects each of the vertices of a triangle. This is known as the circle that circumscribes the triangle.

Key Concepts, continuedIncenters

• The angle bisectors of a triangle are rays that cut the measure of each vertex in half.

• The three angle bisectors of a triangle are also concurrent.

• This point of concurrency is called the incenter of the triangle.

• The incenter of a triangle is equidistant from the sides of the triangle. This is known as the Incenter Theorem.

Theorem

Incenter Theorem The incenter of a triangle is equidistant from the sides of a triangle.

The incenter of this triangle is at X.

Key Concepts, continued• The incenter is always inside the triangle.

Key Concepts, continued• The incenter of a triangle is the center of the circle

that connects each of the sides of a triangle. This is known as the circle that inscribes the triangle.

Key Concepts, continuedOrthocenters

• The altitudes of a triangle are the perpendicular lines from each vertex of the triangle to its opposite side, also called the height of the triangle.

• The three altitudes of a triangle are also concurrent.

• This point of concurrency is called the orthocenter of the triangle.

Key Concepts, continued• The orthocenter can be inside the triangle, outside the

triangle, or even on the triangle depending on the type of triangle.

• The orthocenter is inside acute triangles, outside obtuse triangles, and at the vertex of the right angle of right triangles.

Key Concepts, continued• Look at the placement of the orthocenter, point X, in

the following examples.

X is inside the triangle.

X is outside the triangle.

X is at the vertex of the right angle.

Key Concepts, continuedCentroids

• The medians of a triangle are segments that join the vertices of the triangle to the midpoint of the opposite sides.

• Every triangle has three medians.

• The three medians of a triangle are also concurrent.

Key Concepts, continued• This point of concurrency is called the centroid of the

triangle.

• The centroid is always located inside the triangle

the distance from each vertex to the midpoint of the

opposite side. This is known as the Centroid

Theorem.

Theorem

Centroid Theorem The centroid of a triangle isthe distance from each vertex to the midpoint of the opposite side.

The centroid of this triangle is at point X.

Key Concepts, continued• The centroid is always located inside the triangle.

• The centroid is also called the center of gravity of a triangle because the triangle will always balance at this point.

Key Concepts, continued• Each point of concurrency discussed is considered a

center of the triangle.

• Each center serves its own purpose in design, planning, and construction.

Center of triangle Intersection of…

Circumcenter Perpendicular bisectors

Incenter Angle bisectors

Orthocenter Altitudes

Centroid Medians

Common Errors/Misconceptions• not recognizing that the circumcenter and orthocenter

are outside of obtuse triangles • incorrectly assuming that the perpendicular bisector of

the side of a triangle will pass through the opposite vertex

• interchanging circumcenter, incenter, orthocenter, and centroid

• confusing medians with midsegments • misidentifying the height of the triangle

Guided Practice

Example 3 has vertices

A (–2, 4), B (5, 4), and

C (3, –2). Find the

equation of each median

of to verify that

(2, 2) is the centroid

Guided Practice: Example 3, continued

1. Identify known information. has vertices A (–2,4), B (5, 4), and C (3, –2).

The centroid is X (2, 2).

The centroid of a triangle is the intersection of the medians of the triangle.

2. Determine the midpoint of each side of the triangle. Use the midpoint formula to find the midpoint of .

Midpoint formula

Substitute (–2, 4) and (5, 4) for (x1, y1) and (x2, y2).

The midpoint of is .

Simplify.

Guided Practice: Example 3, continuedUse the midpoint formula to find the midpoint of .

The midpoint of is (4, 1).26

Midpoint formula

Substitute (5, 4) and (3, –2) for (x1, y1) and (x2, y2).

Simplify.

Guided Practice: Example 3, continuedUse the midpoint formula to find the midpoint of .

Midpoint formula

Substitute (–2, 4) and (3, –2) for (x1, y1) and (x2, y2).

The midpoint of is .

Simplify.

3. Determine the medians of the triangle.

Find the equation of , which is the line that passes through A and the midpoint of .

Use the slope formula to calculate the slope of .

Slope formula

The slope of is

Simplify.

Guided Practice: Example 3, continuedFind the y-intercept of .

The equation of that passes through A and the midpoint

of is .

Point-slope form of a line

Substitute (–2, 4) for

(x1, y1) and for m.

Simplify.

Guided Practice: Example 3, continuedFind the equation of , which is the line that passes through B and the midpoint of .

Slope formula

Substitute (5, 4) and

for (x1, y1) and (x2, y2).

The slope of is

Simplify.

The equation of that passes through B and the midpoint

of is .

Substitute (5, 4) for

(x1, y1) and for m.

Simplify.

Guided Practice: Example 3, continuedFind the equation of , which is the line that passes through C and the midpoint of .

Slope formula

Substitute (3, –2) and

for (x1, y1) and (x2, y2).

The slope of is

Simplify.

The equation of that passes through C and the midpoint

of is .

Substitute (3, –2) for (x1, y1) and –4 for m.

Simplify.

4. Verify that X (2, 2) is the intersection of the three medians. For (2, 2) to be the intersection of the three medians, the point must satisfy each of the equations:

(2, 2) satisfies the equation of the median from A to the midpoint of .

Equation of the median from A to the midpoint of

Substitute X (2, 2) for (x, y).

Simplify.

(2, 2) satisfies the equation of the median from B to the midpoint of . 41

Equation of the median from B to the midpoint of

Simplify.

(2, 2) satisfies the equation of the median from C to the midpoint of .

Equation of the median from C to the midpoint of

Simplify.

4. State your conclusion. X (2, 2) is the centroid of with vertices A (–2, 4), B (5, 4), and C (3, –2) because X satisfies each of the equations of the medians of the triangle.

Guided Practice

Example 4Using from

Example 3, which has

vertices A (–2, 4),

B (5, 4), and C (3, –2),

verify that the centroid,

X (2, 2), is the

distance from each

vertex. 45

1. Identify the known information. has vertices A (–2, 4), B (5, 4), and C (3, –2).

The centroid is X (2, 2).

The midpoints of are T , U (4, 1), and

2. Use the distance formula to show that

point X (2, 2) is the distance from each

vertex. Use the distance formula to calculate the distance from A to U.

Distance formula

The distance from A to U is units.

Simplify.

Guided Practice: Example 4, continuedCalculate the distance from X to A.

Distance formula

Substitute (2, 2) and (–2, 4) for (x1, y1) and (x2, y2).

The distance from X to A is units.

Simplify.

X is the distance from A.

Centroid Theorem

Substitute the distances found for AU and XA.

Simplify.

Guided Practice: Example 4, continuedUse the distance formula to calculate the distance from B to V.

Distance formula

Substitute (5, 4) and

for (x1, y1) and

(x2, y2).

Simplify.

The distance from B to V is units.

Guided Practice: Example 4, continuedCalculate the distance from X to B.

Distance formula

Substitute (2, 2) and (5, 4) for (x1, y1) and (x2, y2).

The distance from X to B is units.

Simplify.

X is the distance from B.

Centroid Theorem

Substitute the distances found for BV and XB.

Simplify.

Guided Practice: Example 4, continuedUse the distance formula to calculate the distance from C to T.

Distance formula

Substitute (3, –2) and

for (x1, y1) and

(x2, y2).

Simplify.

The distance from C to T is units.

Guided Practice: Example 4, continuedCalculate the distance from X to C.

Distance formula

Substitute (2, 2) and (3, –2) for (x1, y1) and (x2, y2).

The distance from X to C is units.

Simplify.

X is the distance from C.

Centroid Theorem

Substitute the distances found for CT and XC.

Simplify.

The centroid, X (2, 2), is the distance from each

vertex.

Introduction Think about all the properties of triangles we have learned so far and all the...

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