Rectangle and a random point

Geometry Level 3

The above shows a rectangle A B C D ABCD with P P as a point in the interior of this rectangle.

We are given the distances D P = 3 , A P = 5 , B P = 160 DP = 3, AP = 5, BP = \sqrt{160} .

Find the length of C P CP .


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The answer is 12.

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1 solution

Let D E = x , E P = y , D C = p , and A D = l \text{Let} DE = x, \space EP = y, \space DC = p, \space \text{ and } AD = l .

D P E D E 2 + E P 2 = x 2 + y 2 = 9 \triangle DPE \rightarrow DE^2 + EP^2 = x^2 + y^2 = 9

A P F P F 2 + A F 2 = 25 x 2 + ( l y ) 2 = 25 x 2 + y 2 + l 2 2 l y = 25 l 2 2 l y + 9 = 25 l 2 2 l y = 16 \begin{aligned} \triangle APF \rightarrow PF^2 + AF^2 & = 25 \\ x^2 + ( l-y)^2 & = 25 \\ \color{#3D99F6}{x^2 + y^2} \color{#333333}+ l^2 - 2ly & = 25 \\ l^2 -2ly + \color{#3D99F6}{9} \color{#333333} & = 25 \\ \color{#D61F06} l^2 - 2ly & \color{#333333}= 16\end{aligned}

B P H P H 2 + H B 2 = ( p x ) 2 + ( l y ) 2 = 160 ( p x ) 2 + l 2 2 l y + y 2 = 160 ( p x ) 2 + y 2 + 16 = 160 ( p x ) 2 + y 2 = 144 \begin{aligned} \triangle BPH \rightarrow PH^2 + HB^2 & = (p-x)^2+ (l-y)^2 = 160 \\ (p-x)^2 + \color{#D61F06} l^2 - 2ly \color{#333333}+ y^2 & = 160 \\ (p-x)^2 + y^2 + \color{#D61F06} 16 & \color{#333333} =160 \\ \color{#67D94C} (p-x)^2 + y^2 & = 144 \end{aligned}

P C H C P 2 = P H 2 + C H 2 C P 2 = ( p x ) 2 + y 2 C P 2 = 144 C P = 12 \begin{aligned} \triangle PCH \rightarrow CP^2 & = PH^2 + CH^2 \\ CP^2 & = \color{#67D94C} (p-x)^2 + y^2 \\ CP^2 & = \color{#67D94C} 144 \color{#333333} \rightarrow CP = 12 \end{aligned}

Alternative

By British Flag Theorem, A P 2 + C P 2 = B P 2 + D P 2 AP^2 + CP^2 = BP^2 + DP^2

25 + C P 2 = 160 + 9 25 + CP^2 = 160 + 9

C P 2 = 144 C P = 12 CP^2=144 \rightarrow CP= 12 .

Oh, thank you. Honestly, I didn't even know about "British Flag Theorem" until you give a comment. Thank you- @Brian Charlesworth

Fidel Simanjuntak - 4 years, 5 months ago

You've given a nice proof of the British Flag Theorem .

Brian Charlesworth - 4 years, 5 months ago

Yup british flag theorem makes this too easy!

Prakhar Bindal - 4 years, 5 months ago

Thank u @Fidel Simanjuntak for such a nice soln and @Prakhar Bindal for the British Flag theorem!

Toshit Jain - 4 years, 3 months ago

I solve this by let side 3,5 lean at left side to create 3 traingle only, then using Pythagoras theorem to get 12

Kelvin Hong - 3 years, 10 months ago

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