Math Is Fun Forum

My teacher assigned this homework called marble slide parabolas and I need to make it up to slide 16 but it's way too hard and I'm dying trying to figure it out.
Here's the link :
https://teacher.desmos.com/marbleslides-parabolas

Go there create a class code and try the problems...it's really hard. Can someone please give me the equation for all the slides from 4-16. THANK YOU so much!!!

I need help with these 2 problems:

1)Two circles are externally tangent at point $P$, as shown. Segment $\overline{CPD}$ is parallel to common external tangent $\overline{AB}$. Prove that the distance between the midpoints of $\overline{AB}$ and $\overline{CD}$ is $AB/2$.

2)In triangle $ABC$, $AB = AC$, $D$ is the midpoint of $\overline{BC}$, $E$ is the foot of the perpendicular from $D$ to $\overline{AC}$, and $F$ is the midpoint of $\overline{DE}$. Prove that $\overline{AF}$ is perpendicular to $\overline{BE}$.

Can you solve these 2 problems by writing simple geometry proofs? THANK YOU!!

I need help with these problems:

1)Let $\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If the area of $\triangle ABC$ is $150$ and $AC = 25,$ then what is $BD$?

2)A decorative arrangement of floor tiles forms concentric circles, as shown in the figure to the right. The smallest circle has a radius of 2 feet, and each successive circle has a radius 2 feet longer. All the lines shown intersect at the center and form 12 congruent central angles. What is the area of the shaded region? Express your answer in terms of $\pi$.

Picture for number 2 is :

3) Let $ABC$ be a triangle. We construct squares $ABST$ and $ACUV$ with centers $O_1$ and $O_2$, respectively, as shown. Let $M$ be the midpoint of $\overline{BC}$.

(a) Prove that $\overline{BV}$ and $\overline{CT}$ are equal in length and perpendicular.

(b) Prove that $\overline{O_1 M}$ and $\overline{O_2 M}$ are equal in length and perpendicular.

Thank you so much!!