Mathematics
Grade 10
15 min
Identify similar figures
Identify similar figures
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Introduction & Learning Objectives
Learning Objectives
Define geometric similarity for polygons.
Identify corresponding angles and corresponding sides of two polygons.
Verify the two critical conditions for similarity: congruent corresponding angles and proportional corresponding sides.
Calculate the scale factor (ratio of similarity) between two similar figures.
Write a correct similarity statement using proper notation.
Distinguish between similar and congruent figures.
Apply the properties of similarity to determine if two polygons are similar.
Ever tried to resize a photo and it ended up looking stretched and distorted? 🖼️ Understanding similarity is the mathematical secret to scaling things perfectly!
In this tutorial, we will explore the concept of similar figures – figures that have the same shape b...
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Key Concepts & Vocabulary
TermDefinitionExample
Similar PolygonsTwo polygons are considered similar if two conditions are met: 1) all pairs of corresponding angles are congruent, and 2) the lengths of all pairs of corresponding sides are proportional.A 3x4 rectangle is similar to a 6x8 rectangle because all angles are 90° and the sides are proportional (3/6 = 4/8 = 1/2).
Corresponding AnglesAngles that are in the same relative position in two different polygons. In similar figures, these angles are congruent (have the same measure).If quadrilateral ABCD ~ EFGH, then ∠A corresponds to ∠E, ∠B corresponds to ∠F, and so on.
Corresponding SidesSides that are in the same relative position in two different polygons. In similar figures, the ratio of the lengths of these sides is constant.If ΔABC ~ ΔXYZ, then side AB corre...
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Core Formulas
Condition 1: Congruent Corresponding Angles
For two polygons to be similar, all pairs of corresponding angles must be congruent. For polygons ABCDE and FGHIJ, this means: \angle A \cong \angle F, \angle B \cong \angle G, \angle C \cong \angle H, ...
This rule ensures that the two figures have the exact same shape. You must check every pair of corresponding angles.
Condition 2: Proportional Corresponding Sides
For two polygons to be similar, the ratios of the lengths of all pairs of corresponding sides must be equal to a constant scale factor, k. For polygons ABCDE and FGHIJ, this means: \frac{AB}{FG} = \frac{BC}{GH} = \frac{CD}{HI} = ... = k
This rule ensures that one figure is a perfect enlargement or reduction of the other. You must set up and simplify the ratios for all p...
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Challenging
Triangle A is similar to Triangle B with a scale factor of 2. Triangle B is similar to Triangle C with a scale factor of 1.5. What is the relationship between Triangle A and Triangle C?
A.Triangle A is similar to Triangle C with a scale factor of 3.
B.Triangle A is similar to Triangle C with a scale factor of 3.5.
C.Triangle A is not necessarily similar to Triangle C.
D.Triangle A is congruent to Triangle C.
Challenging
A rectangle has length L and width W. A new rectangle is created with length 1.5L and width 1.5W. Are the two rectangles similar?
A.Only if L = W.
B.Yes, always.
C.No, never.
D.Only if L = 1.5W.
Challenging
Quadrilateral ABCD has vertices A(0,0), B(4,0), C(4,2), D(0,2). Quadrilateral EFGH has vertices E(0,0), F(2,0), G(2,1), H(0,1). Are these two figures similar?
A.No, because the angles are not congruent.
B.Yes, because they are both rectangles.
C.Yes, because the ratio of corresponding sides is 2.
D.No, because the ratio of corresponding sides is not constant.
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