Curriculum / Math / 10th Grade / Unit 3: Dilations and Similarity / Lesson 6
Math
Unit 3
10th Grade
Lesson 6 of 18
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Prove that a line parallel to one side of a triangle divides the other two sides proportionally.
The core standards covered in this lesson
G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
The foundational standards covered in this lesson
8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Below is a diagram of two line segments, $${\overline{AE}}$$ and $${\overline{AD}}$$, drawn on a lined sheet of paper.
Two horizontal line segments, $${{\overline{CB}}}$$ and $${{\overline{ED}}}$$, are drawn.
What is the relationship between $${{\overline{CB}}}$$ and $${{\overline{ED}}}$$ ? What is the relationship between $$\overline{AC}, {\overline{AE}}$$ and $$\overline{AB}, {\overline{AD}}$$?
Module 6: Similarity and Right Triangle Trigonometry from Secondary Mathematics Two: An Integrated Approach made available by Mathematics Vision Project under the CC BY 4.0 license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 1:48 p.m..
In the diagram below,
$${AB= \frac{4}{3}AD}$$
$${AC=\frac{4}{3}AE}$$
What can you prove about $${\overline{DE}}$$ and $${\overline{BC}}$$?
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Given that $${\overline{AB'}}$$ is a dilation of  $${\overline{AB}}$$ by a scale factor $$r$$ from point $$A$$ and $$\overline{AC'}$$ is a dilation of $$\overline{AC}$$ by a scale factor $$r$$ from point $$A$$, prove that $$\overline{BC}\parallel\overline{B'C'}$$.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 5
Lesson 7
Topic A: Dilations off the Coordinate Plane
Describe properties of scale drawings.
G.SRT.A.2
Define and describe the characteristics of dilations. Dilate figures using constructions when the center of dilation is not on the figure.Â
G.CO.A.2 G.SRT.A.2 G.SRT.A.3
Verify that dilations result in congruent angles and proportional line segments.
Divide a line segment into equal sections using dilation.Â
G.CO.D.12 G.SRT.A.1.A G.SRT.A.1.B
Dilate a figure from a point on the figure. Use the properties of dilations with respect to parallel lines to verify dilations and find the center of dilation.
G.CO.D.12 G.SRT.A.1.A G.SRT.A.2
G.CO.C.10 G.SRT.B.4 G.SRT.B.5
Identify measurements in dilated figures with the center of dilation on the figure directly and algebraically.
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Topic B: Dilations on the Coordinate Plane
Dilate a figure on the coordinate plane when the center of dilation is the origin.
G.CO.A.2 G.SRT.A.2
Dilate a figure when the center of dilation is not the origin. Determine center of dilation given the original and dilated figure.
Topic C: Defining Similarity
Define similarity transformation as the composition of basic rigid motions and dilations. Describe similarity transformation applied to an arbitrary figure.
G.SRT.A.2 G.SRT.B.5
Prove that all circles are similar.
G.C.A.1
Prove angle-angle criterion for two triangles to be similar.
G.SRT.A.3
Use angle-angle criterion to prove two triangles to be similar.
Develop the side splitter theorem and side-angle-side similarity criteria, and use these in the solution of problems.
G.SRT.B.5
Topic D: Similarity Applications
Develop the angle bisector theorem based on facts about similarity and congruence, and use this in the solution of problems.
Use the side-side-side criteria for similarity and other similarity and congruence theorems in the solution of problems.
Solve for measurements involving right triangles using scale factors and ratios.
Solve real-life problems with two different centers of dilation.
G.SRT.B.4 G.SRT.B.5
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