In acute triangle $abc$, points $d$, $e$, and $f$ are located on sides $\overline{bc}$, $\overline{ac}$, and $\overline{ab}$ so that $\overline{ad} \perp \overline{bc}$, $\overline{de} \perp \overline{ac}$, and $\overline{df} \perp \overline{ab}$. let $r_1$ and $r_2$ be the radii of the circles around $\triangle abc$ and $\triangle aef$, respectively. determine the number of degrees in the measure of $\angle a$ if the area of $\triangle abc$ is equal to $r_1r_2$.

You drop a rubber ball off the roof of a 50 meter high building onto a paved parking lot. it bounces back up with every bounce, but not quite all the way back up to you. after the first bounce it bounces back only 80 percent of the distance it was dropped from. the pattern continues, meaning after every bounce it comes up to just 80 percent of the previous maximum height. so if before the first bounce the height is 50 meters, what height does the ball reach after the fifth bounce? round your answer to one decimal place and chose the correct response from the choices below:

Cora bought a package of erasers. there are 4 pink erasers and 12 blue erasers. what is the ratio of pink erasers to blue erasers?

Do more republicans (group a) than democrats (group b) favor a bill to make it easier for someone to own a firearm? two hundred republicans and two hundred democrats were asked if they favored a bill that made it easier for someone to own a firearm. how would we write the alternative hypothesis?