Mathematics
Grade 8
15 min
Similar and congruent figures
Similar and congruent figures
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1
Introduction & Learning Objectives
Learning Objectives
Define and differentiate between congruent and similar figures.
Identify corresponding angles and corresponding sides in congruent and similar figures.
Determine if two figures are congruent using rigid transformations (translation, rotation, reflection).
Determine if two figures are similar using dilation and scale factor.
Calculate the scale factor between two similar figures.
Use properties of similar figures to find unknown side lengths or angle measures.
Have you ever noticed how a small toy car looks exactly like a real car, just shrunken down? 🚗 Or how two identical puzzle pieces fit perfectly together? 🧩
In this lesson, we'll explore the fascinating world of similar and congruent figures. You'll learn how to identify these figures, un...
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Key Concepts & Vocabulary
TermDefinitionExample
Congruent FiguresFigures that have the exact same size and shape. One can be transformed into the other using only rigid transformations (translation, rotation, reflection).Two identical squares, each with side length 5 cm.
Similar FiguresFigures that have the same shape but can be different sizes. One can be transformed into the other using rigid transformations and a dilation.A small triangle and a larger triangle where all corresponding angles are equal, and corresponding sides are proportional.
Corresponding AnglesAngles that are in the same relative position in two different figures.In two similar triangles ABC and DEF, angle A corresponds to angle D.
Corresponding SidesSides that are in the same relative position in two different figures.In two similar triangle...
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Core Formulas
Condition for Congruence
If Figure A $\cong$ Figure B, then all corresponding angles are equal AND all corresponding side lengths are equal.
Use this rule to determine if two figures are identical in size and shape. You can also use rigid transformations to map one onto the other.
Condition for Similarity
If Figure A $\sim$ Figure B, then all corresponding angles are equal AND all corresponding side lengths are proportional (meaning they have the same scale factor): $\frac{\text{Side 1 of A}}{\text{Side 1 of B}} = \frac{\text{Side 2 of A}}{\text{Side 2 of B}} = \text{scale factor}$.
Use this rule to determine if two figures have the same shape but potentially different sizes. A dilation is involved.
Scale Factor Calculation
$\text{Scale Factor} = \frac{\text{Length of...
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Challenging
Triangle ABC has vertices A(1, 4), B(4, 2), and C(2, 1). Triangle A'B'C' has vertices A'(-4, 1), B'(-2, 4), and C'(-1, 2). What single transformation maps ΔABC to ΔA'B'C'?
A.translation 5 units left and 3 units down
B.rotation of 90 degrees counter-clockwise about the origin
C.reflection across the y-axis
D.dilation with a scale factor of -1
Challenging
ΔJKL is similar to ΔMNO. The length of JK is x + 4, the length of MN is 2x - 1, the length of KL is 15, and the length of NO is 25. Find the length of side JK.
A.27
B.23
C.45
D.15
Challenging
A point P(5, 7) is a vertex on a triangle. The triangle is dilated by a scale factor of 2 with a center of dilation at C(3, 1). What are the coordinates of P', the image of point P?
A.(10, 14)
B.(8, 8)
C.(7, 13)
D.(6, 12)
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