How to approach this question

Let midpoint of $OA$ be $P$ and $Q$ is the point on $BC$ such that $BQ : QC = 3 : 1$

Find the length of $PQ.$

Meanwhile $h = 4 \sqrt7$ cm

[ You are expected to use Pythagorean theorem only ]

Note by Syed Hamza Khalid
3 months, 2 weeks ago

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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:

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

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

Yea I am sorry it is just as you said. I have edited the problem

- 3 months, 2 weeks ago

But still the ans for PQ is $4 \sqrt{5}$ how?

- 3 months, 2 weeks ago

on point!

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

Yep. I'm sorry I mistyped.

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

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.

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

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

- 3 months, 2 weeks ago

not mention, it's my pleasure

- 3 months, 2 weeks ago