Mathematics
Grade 10
15 min
Similar triangles and similarity transformations
Similar triangles and similarity transformations
What you'll learn
- Solve division problems involving numbers ending in zeroes (e.g., 3600 ÷ 60) with at least 80% accuracy.
- Explain the pattern of how the number of zeroes in the dividend and divisor affects the quotient when dividing numbers ending in zeroes, using at least two examples.
- Identify and correct errors in incorrectly solved division problems involving numbers ending in zeroes, providing a written explanation for each correction.
- Apply the strategy of canceling out zeroes to simplify division problems with numbers ending in zeroes (e.g., canceling zeroes in 2400 ÷ 30 to solve 240 ÷ 3), and justify why this strategy works.
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Define similarity in terms of similarity transformations (a composition of rigid motions and a dilation).
Identify and apply the Angle-Angle (AA), Side-Side-Side (SSS), and Side-Angle-Side (SAS) similarity criteria to prove triangles are similar.
Determine the scale factor between two similar triangles.
Use the properties of similar triangles to find unknown side lengths and angle measures.
Write formal geometric proofs involving triangle similarity.
Solve real-world problems by modeling them with similar triangles.
How can you measure the height of a skyscraper or a giant tree without actually climbing it? 🌳 The secret lies in the power of similar triangles!
This tutorial explores the concept of similarity, where figures have the same shape but differe...
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Key Concepts & Vocabulary
TermDefinitionExample
Similarity TransformationA transformation that produces a similar figure. It is a composition of one or more rigid motions (translation, reflection, rotation) and a dilation.Triangle ABC is translated 2 units right and then dilated by a scale factor of 3 from the origin to create a new, larger triangle A'B'C' that is similar to ABC.
DilationA transformation that enlarges or reduces a figure by a specific scale factor about a fixed point called the center of dilation. It preserves angle measures but not side lengths.A triangle with vertices (1,2), (3,1), and (2,4) is dilated by a scale factor of 2 from the origin. The new vertices are (2,4), (6,2), and (4,8).
Scale Factor (k)The ratio of the lengths of two corresponding sides of similar polygons. If k &...
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Core Formulas
Angle-Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
This is the most common and often the easiest way to prove triangles are similar. If you can find two pairs of congruent angles, you're done. This is often used in problems with parallel lines or shared angles.
Side-Side-Side (SSS) Similarity Theorem
If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar. If (AB/DE) = (BC/EF) = (AC/DF), then ΔABC ~ ΔDEF.
Use this when you are given all three side lengths for both triangles but no angle measures. You must check that the ratios of all three pairs of corresponding sides are equal.
Side-Angle-Side (SAS) Similarity The...
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Challenging
In the figure, a person is standing at point C, and ∠BAC ≅ ∠EDC. To prove that ΔABC ~ ΔEDC to find the width of a river (AB), what additional piece of information is required?
A.The length of side BC
B.The measure of angle B
C.No additional information is needed
D.The lengths of sides AB and ED
Challenging
Given ΔABC with side lengths a, b, and c. A second triangle, ΔA'B'C', is formed with side lengths ka, kb, and kc, where k is a positive constant. Which statement is always true?
A.The triangles are similar only if k > 1.
B.The triangles are congruent for any value of k.
C.The triangles are similar only if 0 < k < 1.
D.The triangles are similar by SSS for any positive k.
Challenging
A sequence of transformations is applied to ΔABC. First, a dilation with a scale factor of 3 centered at the origin. Second, a reflection across the y-axis. Third, a dilation with a scale factor of 1/3 centered at the origin. What is the relationship between the final image and the original ΔABC?
A.The final image is congruent to ΔABC.
B.The final image is similar to ΔABC with a scale factor of 9.
C.The final image is similar to ΔABC with a scale factor of 1.
D.The final image is larger than and in a different quadrant than ΔABC.
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Frequently asked questions
What grade level is "Similar triangles and similarity transformations"?
Similar triangles and similarity transformations is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Similar triangles and similarity transformations?
You'll be able to: Solve division problems involving numbers ending in zeroes (e.g., 3600 ÷ 60) with at least 80% accuracy; Explain the pattern of how the number of zeroes in the dividend and divisor affects the quotient when dividing numbers….
Is "Similar triangles and similarity transformations" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Similar triangles and similarity transformations?
This lesson includes 23 practice questions across multiple difficulty levels, each with instant feedback and explanations.