Mathematics
Grade 10
15 min
Similar triangles and similarity transformations
Similar triangles and similarity transformations
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1
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
In the diagram, quadrilateral ABCD is a trapezoid with AB || DC. The diagonals AC and BD intersect at E. Which statement correctly proves that ΔABE ~ ΔCDE?
A.∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE by vertical angles; similarity by AA.
B.∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE by alternate interior angles; similarity by AA.
C.AE/EC = BE/ED by definition of a trapezoid; similarity by SSS.
D.∠AEB ≅ ∠CED by alternate interior angles; similarity by SAS.
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.
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