How to approach this question

Let midpoint of OAOA be PP and QQ is the point on BCBC such that BQ:QC=3:1 BQ : QC = 3 : 1

Find the length of PQ.PQ.

Meanwhile h=47 h = 4 \sqrt7 cm

[ You are expected to use Pythagorean theorem only ]

Note by Syed Hamza Khalid
1 week, 5 days ago

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Let the center of the base square ABCDABCD be the origin (0,0,0)(0,0,0) and A(4,4,0)A(4,-4,0), B(4,4,0)B(4,4,0), C(4,4,0)C(-4,4,0), D(4,4,0)D(-4,-4,0), and O(0,0,h)=O(0,0,47)O(0,0,h) = O(0,0,4\sqrt 7). Note that h=122(42)2=47h = \sqrt{12^2-(4\sqrt 2)^2} = 4\sqrt 7.

Then P(4+02,4+02,0+472)=P(2,2,27)P \left(\frac {4+0}2, \frac {-4+0}2, \frac {0+4\sqrt 7}2\right) = P (2,-2,2\sqrt 7) and Q(2,4,0)Q(-2,4,0). By Pythagorean theorem:

PQ2=(2(2))2+(24)2+(270)2=80    PQ=45\begin{aligned} PQ^2 & = (2-(-2))^2 + (-2-4)^2 + (2\sqrt 7-0)^2 = 80 \\ \implies PQ & = \boxed{4\sqrt 5} \end{aligned}

Chew-Seong Cheong - 1 week, 5 days ago

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Nop. PQ = 45 4 \sqrt{5} but I don't know how

Syed Hamza Khalid - 1 week, 5 days ago

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Note that hh cannot be 12 cm\text{12 cm} because the slanting sides AO=BO=CO=DO=12 cmAO=BO=CO=DO=\text{12 cm}.

Chew-Seong Cheong - 1 week, 5 days ago

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I have got it. h=47h = 4\sqrt 7.

Chew-Seong Cheong - 1 week, 5 days ago

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@Chew-Seong Cheong Yea I am sorry it is just as you said. I have edited the problem

Syed Hamza Khalid - 1 week, 4 days ago

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@Chew-Seong Cheong But still the ans for PQ is 45 4 \sqrt{5} how?

Syed Hamza Khalid - 1 week, 4 days ago

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on point!

nibedan mukherjee - 1 week, 4 days ago

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the problem says that h (*height of the pyramid) = 12 and also OA = OD = OC = OB =12 , which is geometrically not possible.

nibedan mukherjee - 1 week, 4 days ago

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Yep. I'm sorry I mistyped.

Syed Hamza Khalid - 1 week, 4 days ago

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yep I will post the solution with diagram .. but it's not possible until you post this note as a problem

nibedan mukherjee - 1 week, 3 days ago

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The information h=47h=4\sqrt 7 is not necessary for solving the problem. It can be found from other information given. It should be mentioned if it is set as a problem.

Chew-Seong Cheong - 1 week, 3 days ago

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@Chew-Seong Cheong you are absolutely right sir, do check my solution....here" https://brilliant.org/problems/i-dont-know-why-its-not-easy/#!/solution-comments/241234/"

nibedan mukherjee - 1 week, 3 days ago

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Please find PQ

Syed Hamza Khalid - 1 week, 4 days ago

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I have provided the solution in the discussion panel of your problem named " I don't know why its not easy..." geometry level 2

nibedan mukherjee - 1 week, 3 days ago

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@Nibedan Mukherjee Oh okay... I checked its great. I clearly understand. Thank you so much for solving it.

Syed Hamza Khalid - 1 week, 3 days ago

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@Syed Hamza Khalid not mention, it's my pleasure

nibedan mukherjee - 1 week, 3 days ago

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