Geometry Lesson

Zoom In, Zoom Out: Dilations, Scale Factor & Triangle Similarity

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High SchoolThin-SlicingCCSS.MATH.CONTENT.HSG.SRT.A.1, CCSS.MATH.CONTENT.HSG.SRT.A.3, CCSS.MATH.CONTENT.HSG.SRT.B.550 min

Learning Objective

Students will be able to determine whether two triangles are similar using AA, SAS, or SSS similarity criteria, identify the scale factor of a dilation, and apply similarity to solve for unknown side lengths and distances.

Lesson Overview

Students progress from identifying scale factors in dilations to applying AA, SAS, and SSS similarity criteria, culminating in multi-step similarity proofs and indirect measurement problems. Tasks change one variable at a time, building from visual recognition to formal reasoning.

Materials

Vertical whiteboards or chart paper
Dry-erase markers
Rulers or measuring tape
Random grouping tool
Printed Four-Quadrant note sheets

Scaffolded Task Progression

1Task 1

Triangle $ABC$ has vertices $A(0,0)$, $B(4,0)$, and $C(0,3)$. Triangle $A'B'C'$ has vertices $A'(0,0)$, $B'(8,0)$, and $C'(0,6)$. 1. Plot both triangles on the coordinate plane. 2. Find the ratio $\frac{A'B'}{AB}$, $\frac{A'C'}{AC}$, and $\frac{B'C'}{BC}$. 3. What single number describes how triangle $ABC$ was resized? ```coordgraph {"xRange":[0,10],"yRange":[0,8],"xStep":1,"yStep":1,"xLabel":"x","yLabel":"y","polylines":[{"points":[[0,0],[4,0],[0,3],[0,0]],"color":"#2563eb"},{"points":[[0,0],[8,0],[0,6],[0,0]],"color":"#dc2626"}],"points":[{"x":0,"y":0,"label":"A=A'"},{"x":4,"y":0,"label":"B"},{"x":0,"y":3,"label":"C"},{"x":8,"y":0,"label":"B'"},{"x":0,"y":6,"label":"C'"}]} ```

2Task 2

Triangle

P QR

has sides

P Q = 6

QR = 8

P R = 10

. Triangle P'Q'R' has sides

P^{'} Q^{'} = 9

Q^{'} R^{'} = 12

P^{'} R^{'} = 15

.
1. Find the ratio of each pair of corresponding sides.
2. What is the scale factor from

△ P QR

△ P^{'} Q^{'} R^{'}

?
3. What is the scale factor from

△ P^{'} Q^{'} R^{'}

back to

△ P QR

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