Given a rectangle $ABCD$ with a point $P$ inside it. It is known that $PA = 17, PB = 15,$ and $PC = 6.$ What is the length of $PD$?
Problem
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Tags: rectangle, geometry
Socoobo
19.09.2020 02:45
Let the distances from $P$ to each of the sides be $a,b,c,d$. We have
$$a^2+b^2=289$$$$b^2+c^2=225$$$$c^2+d^2=36$$
Add the first and third equations and subtract by the second. This gives
$$a^2+d^2=100$$
so the answer is $\boxed{10}$.
ilovepizza2020
19.09.2020 02:46
Socoobo wrote:
Let the distances from $P$ to each of the sides be $a,b,c,d$. We have
$$a^2+b^2=289$$$$b^2+c^2=225$$$$c^2+d^2=36$$
Add the first and third equations and subtract by the second. This gives
$$a^2+d^2=100$$
so the answer is $\boxed{10}$.
FTFY
franzliszt
19.09.2020 02:50
By British Flag Theorem, we have $PA^2+PC^2=PB^2+PD^2 \iff 100=PD^2$ so $PD=10$. Nice!
ilovepizza2020
19.09.2020 02:51
What's the British Flag Theorem? Also, hide your sols.
Socoobo
19.09.2020 02:51
franzliszt wrote: By British Flag Theorem, we have $PA^2+PC^2=PB^2+PD^2 \iff 100=PD^2$ so $PD=10$. Nice! I cannot believe I have done competition math since like 6th grade and have never heard of this theorem... EDIT: @above after looking it up on google, it's supposed to be a pretty well known theorem and it would've been pretty helpful for mathcounts but unfortunately I'm in 11th grade already so...
franzliszt
19.09.2020 02:52
I never thought I would get to use it!!
GammaZero
19.09.2020 03:05
People who are interested can derive the British Flag Theorem by using the Pythagorean Theorem
Iamnobody
19.09.2020 06:03
https://artofproblemsolving.com/wiki/index.php/British_Flag_Theorem Funny story. Two years ago, I was doing this problem, and I happened to remember this theorem.
ApraTrip
19.09.2020 06:20
Wait isn't the "british flag theorem" just the perpendicularity lemma restated?
Iamnobody
19.09.2020 06:26
ApraTrip wrote: Wait isn't the "british flag theorem" just the perpendicularity lemma restated? I think so, but it sounds cooler.