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9-12 Integrated Math 1 Unit 5 Lesson 10

Gradeband

K-5 Math 6-8 Math •9-12 Math

Course

•Integrated Math 1 Integrated Math 2 Integrated Math 3

Unit

Unit 1 Unit 2 Unit 3 Unit 4 •Unit 5 Unit 6 Unit 7 Unit 8 Unit 9

Modeling Prompts

Lesson

Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 •Lesson 10 Lesson 11

K-5
6-8
9-12

Integrated Math 1
Integrated Math 2
Integrated Math 3

Algebra 1
Geometry
Algebra 2
Extra Support Materials for Algebra 1

Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6
Unit 7
Unit 8
Unit 9

Modeling Prompts

Lesson 1
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11

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Lesson 10

10.1 Warm-up 10.2 Activity 10.3 Activity 10.4 Activity Student Lesson Summary

Unit 5, Lesson 10

Lines in Triangles

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Integrated Math 1

10.1

Warm-up

Folding Altitudes

Draw a triangle on tracing paper. Fold the altitude from each vertex.

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10.2

Activity

Altitude Attributes

Triangle is graphed.

<p>Triangle ABC graphed. A = origin, B = 8 comma 0, C = 2 comma 10<br>
</p>

Find the slope of each side of the triangle.
Find the slope of each altitude of the triangle.
Sketch the altitudes. Label the point of intersection, .
Write equations for all 3 altitudes.
Use the equations to find the coordinates of , and verify algebraically that the altitudes all intersect at .

Are You Ready for More?

10.3

Activity

Percolating on Perpendicular Bisectors

Standards Alignment

Building On

Addressing

G-CO.10

Prove theorems about triangles.

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G-GPE.4

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

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Building Toward

Triangle is graphed.

<p>Triangle ABC graphed. A = origin, B = 8 comma 0, C = 2 comma 10<br>
</p>

Find the midpoint of each side of the triangle.
Sketch the perpendicular bisectors, using an index card to help draw 90-degree angles. Label the intersection point as .
Write equations for all 3 perpendicular bisectors.
Use the equations to find the coordinates of , and verify algebraically that the perpendicular bisectors all intersect at .

Are You Ready for More?

10.4

Activity

Tiling the (Coordinate) Plane

Standards Alignment

Building On

Addressing

G-GPE.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

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Building Toward

A tessellation covers the entire plane with shapes that do not overlap or leave gaps.

Tile the plane with congruent rectangles:
1. Draw the rectangles on your grid.
2. Write the equations for lines that outline 1 rectangle.
Tile the plane with congruent right triangles:
1. Draw the right triangles on your grid.
2. Write the equations for lines that outline 1 right triangle.
Tile the plane with any other shapes:
1. Draw the shapes on your grid.
2. Write the equations for lines that outline 1 of the shapes.

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Student Lesson Summary

The three perpendicular bisectors of a triangle always intersect in one point. We can use coordinate geometry to show that the altitudes of a triangle intersect in one point, too. The three altitudes of triangle are shown here. They appear to intersect at the point . By finding their equations, we can prove this is true.

<p>Triangle ABC graphed. A = origin, B = 4 comma 8, C = 16 comma 0. 3 medians of triangle graphed. medians intersect at 4 comma 6. </p>

The slopes of sides and are 0, , and 2. The altitude from is the vertical line . The slope of the altitude from is . Since the altitude goes through its equation is . The slope of the altitude from is . Following this slope over to the -axis we can see that the -intercept is 8. So the equation for this altitude is .

We can now verify that lies on all three altitudes by showing that the point satisfies the three equations. By substitution, we see that each equation is true when and .

Glossary

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Standards Alignment

Building On

G-CO.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

View Full Standard

Addressing

Building Toward

G-CO.10

Prove theorems about triangles.

View Full Standard

Standards Alignment

Building On

G-GPE.5

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

View Full Standard

Addressing

G-CO.10

Prove theorems about triangles.

View Full Standard

G-GPE.4

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

View Full Standard

Building Toward