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.
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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.
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
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.
4
1.9.4: Proving Centers of Triangles
Key Concepts, continued
5
1.9.4: Proving Centers of Triangles
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.
6
1.9.4: Proving Centers of Triangles
Key Concepts, continued• Look at the placement of the circumcenter, point X, in
the following examples.
7
1.9.4: Proving Centers of Triangles
Acute triangle Obtuse triangle Right triangle
X is inside the triangle.
X is outside the triangle.
X is on the midpoint of the hypotenuse.
8
1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
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.
Key Concepts, continued
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1.9.4: Proving Centers of Triangles
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.
11
1.9.4: Proving Centers of Triangles
Acute triangle Obtuse triangle Right 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.
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1.9.4: Proving Centers of Triangles
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
Acute triangle Obtuse triangle Right triangle
X is inside the triangle.
X is outside the triangle.
X is at the vertex of the right angle.
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
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.
Key Concepts, continued
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1.9.4: Proving Centers of Triangles
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.
19
1.9.4: Proving Centers of Triangles
Acute triangle Obtuse triangle Right triangle
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.
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1.9.4: Proving Centers of Triangles
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
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1.9.4: Proving Centers of Triangles
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
of .
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 3, continued
2. Determine the midpoint of each side of the triangle. Use the midpoint formula to find the midpoint of .
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1.9.4: Proving Centers of Triangles
Midpoint formula
Substitute (–2, 4) and (5, 4) for (x1, y1) and (x2, y2).
Guided Practice: Example 3, continued
The midpoint of is .
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 3, continuedUse the midpoint formula to find the midpoint of .
The midpoint of is (4, 1).26
1.9.4: Proving Centers of Triangles
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 .
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1.9.4: Proving Centers of Triangles
Midpoint formula
Substitute (–2, 4) and (3, –2) for (x1, y1) and (x2, y2).
Guided Practice: Example 3, continued
The midpoint of is .
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 3, continued
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 3, continued
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 .
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1.9.4: Proving Centers of Triangles
Slope formula
Substitute (–2, 4) and (4, 1) for (x1, y1) and (x2, y2).
Guided Practice: Example 3, continued
The slope of is
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1.9.4: Proving Centers of Triangles
Simplify.
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1.9.4: Proving Centers of Triangles
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 .
Use the slope formula to calculate the slope of .
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1.9.4: Proving Centers of Triangles
Slope formula
Substitute (5, 4) and
for (x1, y1) and (x2, y2).
Guided Practice: Example 3, continued
The slope of is
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 3, continuedFind the y-intercept of .
The equation of that passes through B and the midpoint
of is .
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1.9.4: Proving Centers of Triangles
Point-slope form of a line
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 .
Use the slope formula to calculate the slope of .
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1.9.4: Proving Centers of Triangles
Slope formula
Substitute (3, –2) and
for (x1, y1) and (x2, y2).
Guided Practice: Example 3, continued
The slope of is
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 3, continuedFind the y-intercept of .
The equation of that passes through C and the midpoint
of is .
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1.9.4: Proving Centers of Triangles
Point-slope form of a line
Substitute (3, –2) for (x1, y1) and –4 for m.
Simplify.
Guided Practice: Example 3, continued
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:
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from A to the midpoint of .
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1.9.4: Proving Centers of Triangles
Equation of the median from A to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify.
Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from B to the midpoint of . 41
1.9.4: Proving Centers of Triangles
Equation of the median from B to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify.
Guided Practice: Example 3, continued
(2, 2) satisfies the equation of the median from C to the midpoint of .
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1.9.4: Proving Centers of Triangles
Equation of the median from C to the midpoint of
Substitute X (2, 2) for (x, y).
Simplify.
Guided Practice: Example 3, continued
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.
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1.9.4: Proving Centers of Triangles
✔
Guided Practice: Example 3, continued
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1.9.4: Proving Centers of Triangles
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
1.9.4: Proving Centers of Triangles
Guided Practice: Example 4, continued
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
V .
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 4, continued
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.
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1.9.4: Proving Centers of Triangles
Distance formula
Guided Practice: Example 4, continued
The distance from A to U is units.
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1.9.4: Proving Centers of Triangles
Substitute (–2, 4) and (4, 1) for (x1, y1) and (x2, y2).
Simplify.
Guided Practice: Example 4, continuedCalculate the distance from X to A.
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1.9.4: Proving Centers of Triangles
Distance formula
Substitute (2, 2) and (–2, 4) for (x1, y1) and (x2, y2).
Guided Practice: Example 4, continued
The distance from X to A is units.
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 4, continued
X is the distance from A.
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
Distance formula
Substitute (5, 4) and
for (x1, y1) and
(x2, y2).
Simplify.
Guided Practice: Example 4, continued
The distance from B to V is units.
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 4, continuedCalculate the distance from X to B.
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1.9.4: Proving Centers of Triangles
Distance formula
Substitute (2, 2) and (5, 4) for (x1, y1) and (x2, y2).
Guided Practice: Example 4, continued
The distance from X to B is units.
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 4, continued
X is the distance from B.
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1.9.4: Proving Centers of Triangles
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.
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1.9.4: Proving Centers of Triangles
Distance formula
Substitute (3, –2) and
for (x1, y1) and
(x2, y2).
Simplify.
Guided Practice: Example 4, continued
The distance from C to T is units.
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1.9.4: Proving Centers of Triangles
Guided Practice: Example 4, continuedCalculate the distance from X to C.
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1.9.4: Proving Centers of Triangles
Distance formula
Substitute (2, 2) and (3, –2) for (x1, y1) and (x2, y2).
Guided Practice: Example 4, continued
The distance from X to C is units.
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1.9.4: Proving Centers of Triangles
Simplify.
Guided Practice: Example 4, continued
X is the distance from C.
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1.9.4: Proving Centers of Triangles
Centroid Theorem
Substitute the distances found for CT and XC.
Simplify.
Guided Practice: Example 4, continued
The centroid, X (2, 2), is the distance from each
vertex.
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1.9.4: Proving Centers of Triangles
✔
Guided Practice: Example 4, continued
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1.9.4: Proving Centers of Triangles
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