Question

1. verify that each pair of triangles is similar.

a.
b.

Explanation:

Step1: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate side - lengths of the first pair of triangles (a)

For $\triangle ABC$ with $A(2,3)$, $B(2,5)$, $C(6,3)$:
$AB=\sqrt{(2 - 2)^2+(5 - 3)^2}=\sqrt{0 + 4}=2$;
$BC=\sqrt{(6 - 2)^2+(3 - 5)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$;
$AC=\sqrt{(6 - 2)^2+(3 - 3)^2}=\sqrt{16+0}=4$.
For $\triangle DEF$ with $D(4,6)$, $E(4,10)$, $F(12,6)$:
$DE=\sqrt{(4 - 4)^2+(10 - 6)^2}=\sqrt{0 + 16}=4$;
$EF=\sqrt{(12 - 4)^2+(6 - 10)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}$;
$DF=\sqrt{(12 - 4)^2+(6 - 6)^2}=\sqrt{64+0}=8$.

Step3: Check the ratio of corresponding sides

$\frac{AB}{DE}=\frac{2}{4}=\frac{1}{2}$; $\frac{BC}{EF}=\frac{2\sqrt{5}}{4\sqrt{5}}=\frac{1}{2}$; $\frac{AC}{DF}=\frac{4}{8}=\frac{1}{2}$.
Since the ratios of the corresponding sides of $\triangle ABC$ and $\triangle DEF$ are equal, the two triangles are similar.

Step4: Calculate side - lengths of the second pair of triangles (b)

For $\triangle DEF$ with $D(4,5)$, $E(6,9)$, $F(8,5)$:
$DE=\sqrt{(6 - 4)^2+(9 - 5)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$;
$EF=\sqrt{(8 - 6)^2+(5 - 9)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$;
$DF=\sqrt{(8 - 4)^2+(5 - 5)^2}=\sqrt{16+0}=4$.
For $\triangle ABC$ with $A(8,10)$, $B(12,18)$, $C(16,10)$:
$AB=\sqrt{(12 - 8)^2+(18 - 10)^2}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}$;
$BC=\sqrt{(16 - 12)^2+(10 - 18)^2}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}$;
$AC=\sqrt{(16 - 8)^2+(10 - 10)^2}=\sqrt{64+0}=8$.

Step5: Check the ratio of corresponding sides

$\frac{DE}{AB}=\frac{2\sqrt{5}}{4\sqrt{5}}=\frac{1}{2}$; $\frac{EF}{BC}=\frac{2\sqrt{5}}{4\sqrt{5}}=\frac{1}{2}$; $\frac{DF}{AC}=\frac{4}{8}=\frac{1}{2}$.
Since the ratios of the corresponding sides of $\triangle DEF$ and $\triangle ABC$ are equal, the two triangles are similar.

Answer:

The triangles in part (a) and part (b) are similar.