Question

if $\triangle xyz$ is dilated by a factor of 3, what would be the area of the $\triangle xyz$?
what is the scale factor from $\triangle tuv$ to $\triangle qrs$?

Explanation:

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First Question (Area of Dilated Triangle)

Step1: Find base of △XYZ

From the grid, base $YZ = 2$ units.

Step2: Find height of △XYZ

From the grid, height (vertical distance from X to YZ) $= 3$ units.

Step3: Calculate area of △XYZ

Use triangle area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$
$\text{Area of } \triangle XYZ = \frac{1}{2} \times 2 \times 3 = 3$

Step4: Scale area for dilation

When dilated by scale factor $k$, area scales by $k^2$. Here $k=3$, so $k^2=9$.
$\text{Area of } \triangle X'Y'Z' = 3 \times 9 = 27$

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Second Question (Scale Factor)

Step1: Pick corresponding side lengths

Take side $TU$ (from $\triangle TUV$): distance between $T(-2,0)$ and $U(0,3)$ is $\sqrt{(0-(-2))^2 + (3-0)^2} = \sqrt{4+9}=\sqrt{13}$.
Take side $QR$ (from $\triangle QRS$): distance between $Q(-4,0)$ and $R(0,6)$ is $\sqrt{(0-(-4))^2 + (6-0)^2} = \sqrt{16+36}=\sqrt{52}=2\sqrt{13}$.

Step2: Calculate scale factor

Scale factor = $\frac{\text{Length of } QR}{\text{Length of } TU} = \frac{2\sqrt{13}}{\sqrt{13}} = 2$

Answer:

1. Area of $\triangle X'Y'Z'$: 27
2. Scale factor from $\triangle TUV$ to $\triangle QRS$: 2