LEAP 2025 Geometry Major Content. Practice it free.

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

A reflection is a rigid transformation that maps the points from a preimage, over a line of reflection, to the points of an image that are equidistant to the line. In this example, that line is the perpendicular bisector of a line segment from a point to its image, so the segment maps onto itself.

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 $x = 1{,}250 \times 48 = 60{,}000\text{ people}\textsf{.}$

In a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, the hypotenuse is twice the short leg: $2 \times 10 = 20\textsf{.}$

Both legs of a $45$-$45$-$90$ triangle are equal, so the other leg is $15$.

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.

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

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 $y = -x + 2\textsf{.}$