Proving the Midsegment of a...
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Proving the Midsegment of a TriangleAdapted from Walch Education
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Key Concepts
• In the diagram , the midpoint
of is X.
• The midpoint of is Y.
• A midsegment of is .
21.9.3: Proving the Midsegment of a Triangle
AB
BC
XY
• The midpoint is the point on a line segment that divides the segment into two equal parts.
• A midsegment of a triangle is a line segment that joins the midpoints of two sides of a triangle.
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Key Concepts, continued
• The midsegment of a triangle is parallel to the
third side of the triangle and is half as long as the
third side. This is known as the Triangle
Midsegment Theorem.
• Every triangle has three midsegments.
31.9.3: Proving the Midsegment of a Triangle
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41.9.3: Proving the Midsegment of a Triangle
Theorem
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to the third
side and is half as long.
XY =
1
2AC
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Key Concepts, continued
• When all three of the midsegments of a triangle are connected, a midsegment triangle is created.
51.9.3: Proving the Midsegment of a Triangle
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Key Concepts, continued
• Coordinate proofs, proofs that involve calculations and make reference to the coordinate plane, are often used to prove many theorems.
61.9.3: Proving the Midsegment of a Triangle
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Practice
The midpoints of a triangle are X (–2, 5), Y (3, 1), and Z (4, 8). Find the coordinates of the vertices of the triangle.
71.9.3: Proving the Midsegment of a Triangle
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Plot the midpoints on a coordinate plane.
81.9.3: Proving the Midsegment of a Triangle
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Connect the midpoints to form the midsegments
91.9.3: Proving the Midsegment of a Triangle
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Calculate the slope of each midsegment
Calculate the slope of .
The slope of is .
101.9.3: Proving the Midsegment of a Triangle
Slope formula
Substitute (–2, 5) and (3, 1)
for (x1, y1) and (x2, y2).
Simplify.
XY
XY -
4
5
m =
y2
- y1
x2
- x1
m =
(1) - (5)
(3) - (-2)
m = -
4
5
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Calculate the slope of each midsegmentCalculate the slope of .
The slope of is 7.
111.9.3: Proving the Midsegment of a Triangle
Slope formula
Substitute (3, 1) and (4, 8) for
(x1, y1) and (x2, y2).
Simplify.
YZ
m =
y2
- y1
x2
- x1
m =
(8) - (1)
(4) - (3)
m =
7
1= 7
YZ
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Calculate the slope of each midsegment
Calculate the slope of .
The slope of is .
121.9.3: Proving the Midsegment of a Triangle
Slope formula
Substitute (–2, 5) and (4, 8)
for (x1, y1) and (x2, y2).
Simplify.
XZ
m =
y2
- y1
x2
- x1
m =
(8) - (5)
(4) - (-2)
m =
3
6=
1
2
XZ
1
2
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Draw the lines that contain the midpoints.
The endpoints of each midsegment are the midpoints of the larger triangle.
Each midsegment is also parallel to the opposite side.
131.9.3: Proving the Midsegment of a Triangle
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Draw the lines that contain the midpoints.
The slope of is .
From point Y, draw a line that has a slope of .
141.9.3: Proving the Midsegment of a Triangle
XZ
1
2
1
2
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Draw the lines that contain the midpoints.
The slope of is 7.
From point X, draw a line that has a slope of 7.
151.9.3: Proving the Midsegment of a Triangle
YZ
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Draw the lines that contain the midpoints.
The slope of is .
From point Z, draw a line that has a slope of .
The intersections of the
lines form the vertices
of the triangle.
161.9.3: Proving the Midsegment of a Triangle
XY
-
4
5
-
4
5
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Determine the vertices of the triangle.
The vertices of the triangle are (–3, –2), (9, 4), and (–1, 12), as shown on the following slide.
171.9.3: Proving the Midsegment of a Triangle
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181.9.3: Proving the Midsegment of a Triangle
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Thanks for Watching!
Dr. Dambreville