Question 544·Hard·Math

No Desmos needed. A familiar right-triangle pattern can be checked against both given lengths. In the figure, $AC=13$ is across from the right angle, so it's the hypotenuse. The eastward leg $AB$ touches angle $x$, so it's the adjacent leg. The northward leg $BC$ is across from $x$.

A Pythagorean triple is three whole-number side lengths that make a right triangle. The $5$-$12$-$13$ triple fits the 13-kilometer hypotenuse: $5^2+12^2=13^2$. So $5$ and $12$ are candidate legs. The hypotenuse alone doesn't tell you which leg goes east.

Check the leg difference before assigning the eastward leg: $12-5=7$. The site is farther east than north, so $AB=12$ kilometers and $BC=5$ kilometers.

Cosine means adjacent leg divided by hypotenuse. From angle $x$, that's the eastward leg divided by the direct route:

So the cosine of the route's angle with east is $\frac{12}{13}$. Choice B.

If the whole-number pattern doesn't come to mind, let $n$ be the northward distance. The site is $7$ kilometers farther east, so the eastward distance is $n+7$ kilometers.

East and north meet at a right angle, so the Pythagorean theorem says the squares of the legs add to the square of the direct route:

Type $n^2+(n+7)^2\sim13^2\{n>0\}$ in Desmos. The tilde asks Desmos to find $n$, and the restriction keeps the distance positive. Under PARAMETERS, Desmos shows $n=5$, so the northward distance is $5$ kilometers.

The eastward leg is $n+7$ and is adjacent to $x$, so type $\frac{n+7}{13}$ beneath the regression. Desmos shows $0.923076\ldots$; its fraction button gives $\frac{12}{13}$. So the cosine of the angle between the direct route and east is $\frac{12}{13}$. Choice B.