In the mini-question, does choice iv.) not work?
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Module 2 Week 2 Day 7 Challenge Part 4
In the mini-question, does choice iv. not work? I think it does...
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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
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There's a difference?
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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 \(\text{\color{red}{wrong}}.\) -
ooooooooh thanks
