In the mini-question, does choice iv.) not work?

[The Blade Dancer]

Module 2 Week 2 Day 7 Challenge Part 4

In the mini-question, does choice iv. not work? I think it does...

[nastya]

Hi @The-Rogue-Blade!

In the mini-question, the answer choice $\text{IV}$ will work if we change it to: "If a point cuts a chord in half, then the Power of a Point formula gives us: ... iv) that the distance from the point to the center of the circle gets smaller as $\overline{AB}$ gets bigger". But in our answer choice $\text{IV}$ we have smaller instead, which makes this statement wrong. Just imagine how you move this chord and think of it one more time, and then you'll get it

[The Blade Dancer]

There's a difference?

[nastya]

Well, obviously, there is
Let's take a closer look together.

Here we have our chord $\overline{AB}$ with its midpoint $P$ and a circle with its center $O:$

All points that have distance to the center of our circle that is smaller or equal to the distance from point $P$ to the center $O$ are points in the smaller purple circle:

So, all the chords $\overline{A_iB_i}$ have their midpoint $P_i$ not further than point $P$ from the center $O$ (see diagram below). To the contrary, all the chords $\overline{C_iD_i}$ have their midpoint $R_i$ further than point $P$ to the center $O:$

As we can see from these diagrams, all the $\overline{A_iB_i}$ chords are bigger than the chords $\overline{C_iD_i}.$ So the $\text{\color{blue}\it{distance}}$ from the middle point to the center of the circle gets $\text{\color{blue}\bf{smaller}}$ as the $\text{\color{blue}\it{chord}}$ gets $\text{\color{blue}\bf{bigger}}$, while when the $\text{\color{green}\it{chord}}$ gets $\text{\color{green}\bf{smaller}}$, the $\text{\color{green}\it{distance}}$ gets $\text{\color{green}\bf{bigger}}$. That is why the statement "the $\text{\color{darkred}\it{distance}}$ gets $\text{\color{darkred}\bf{smaller}}$ as $\text{\color{darkred}\it{chord}}$ gets $\text{\color{darkred}\bf{smaller}}$" is $\text{\color{red}{wrong}}.$

[The Blade Dancer]

ooooooooh thanks