Question

nameunderstand and recognize similar figures1. use tools use a coordinate plane to design the background image for a video game. start with △abc.a. to draw △def, translate △abc using (x, y) → (x − 2, y). then apply a dilation with scale factor 2 and center of dilation (0, 0). draw and label △def.b. to draw △ghj, reflect △abc using (x, y) → (x, −y). then apply a dilation with scale factor 2 and center of dilation (0, 0). draw and label △ghj.c. critique reasoning a student claims that all three triangles are similar. do you agree or disagree? explain.

Explanation:

First, identify the coordinates of $\triangle ABC$ from the graph:
$A=(2,2)$, $B=(2,0)$, $C=(0,0)$

Step1: Find $\triangle DEF$ coordinates

First translate, then dilate.
Translation: $(x,y)\to(x-2,y)$
$A'(2-2,2)=(0,2)$, $B'(2-2,0)=(0,0)$, $C'(0-2,0)=(-2,0)$
Dilation (scale 2, center $(0,0)$): $(x,y)\to(2x,2y)$
$D=(0\times2,2\times2)=(0,4)$, $E=(0\times2,0\times2)=(0,0)$, $F=(-2\times2,0\times2)=(-4,0)$

Step2: Find $\triangle GHJ$ coordinates

First reflect, then dilate.
Reflection: $(x,y)\to(x,-y)$
$A''=(2,-2)$, $B''=(2,0)$, $C''=(0,0)$
Dilation (scale 2, center $(0,0)$): $(x,y)\to(2x,2y)$
$G=(2\times2,-2\times2)=(4,-4)$, $H=(2\times2,0\times2)=(4,0)$, $J=(0\times2,0\times2)=(0,0)$

Step3: Verify similarity

Similar figures have proportional sides and equal angles.

• $\triangle ABC$ side lengths: $AB=2$, $BC=2$, $AC=2\sqrt{2}$
• $\triangle DEF$ side lengths: $DE=4$, $EF=4$, $DF=4\sqrt{2}$ (ratio 2:1 to $\triangle ABC$)
• $\triangle GHJ$ side lengths: $GH=4$, $HJ=4$, $GJ=4\sqrt{2}$ (ratio 2:1 to $\triangle ABC$)

All triangles have 45-45-90 angle measures, so corresponding angles are equal.

Answer:

A. Coordinates of $\triangle DEF$: $D=(0,4)$, $E=(0,0)$, $F=(-4,0)$ (plot these points and connect to form the triangle)
B. Coordinates of $\triangle GHJ$: $G=(4,-4)$, $H=(4,0)$, $J=(0,0)$ (plot these points and connect to form the triangle)
C. Agree. All three triangles have congruent corresponding angles (45-45-90) and their corresponding side lengths are in a constant proportional ratio ($\triangle DEF$ and $\triangle GHJ$ are both scaled by a factor of 2 from $\triangle ABC$), so they are all similar.