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

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

The information $h=4\sqrt 7$ 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
–
you are absolutely right sir, do check my solution....here" https://brilliant.org/problems/i-dont-know-why-its-not-easy/#!/solution-comments/241234/"

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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## Comments

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

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

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

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

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

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

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$4 \sqrt{5}$ how?

But still the ans for PQ isLog in to reply

on point!

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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.

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Yep. I'm sorry I mistyped.

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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

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The information $h=4\sqrt 7$ 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.

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Please find PQ

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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

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