LEAP 2025 Geometry Major Content. Practice it free.

Both legs of a $45$-$45$-$90$ triangle are equal, so the other leg is $20$.

Set up the proportion $\dfrac{1{,}250\text{ people}}{1\text{ square mile}} = \dfrac{x\text{ people}}{48\text{ square miles}}\textsf{.}$ Cross-multiplying gives $x = 1{,}250 \times 48 = 60{,}000\text{ people}\textsf{.}$

Same-side interior angles are supplementary: $(6x-5)+(4x+15)=180$, so $10x+10=180$ and $x=17$.

The center is $(4, -5)$ and the radius is $6\textsf{.}$ A point on the circle must satisfy $4 - 6 \le a \le 4 + 6\textsf{,}$ so $-2 \le a \le 10\textsf{.}$ The value $3$ is in this interval.

$y$-coordinate $= \dfrac{8 + 4}{2} = \dfrac{12}{2} = 6$.

Parallel lines have the same slope, so $m = -1$. Substitute $(4, -2)$ into $y = -x + b$: $-2 = -4 + b$, so $b = 2$. The equation is $y = -x + 2\textsf{.}$

Parallel lines have the same slope. In $y = \dfrac{3}{4}x + 8$, the slope is $\dfrac{3}{4}$, so the slope of line $n$ is also $\dfrac{3}{4}\textsf{.}$

Similar figures have a constant scale factor between corresponding lengths. Perimeter ratio $3 : 5$ means each side of the larger triangle is $\dfrac{5}{3}$ times the matching side of the smaller. The larger shortest side is $6 \cdot \dfrac{5}{3} = 10$.