Uninterested in drawing figures

Geometry Level 4

I haven't drawn a trapezoid for this question. I apologize for that.

Anyways, the trapezoid A B C D ABCD has A B = 3 AB = 3 , B C = 10 BC = 10 , C D = 12 CD = 12 , D A = 5 DA = 5

Find B D BD if the measurement of B D BD is an integer.

This is part of the series called " It's easy, believe me! "


The answer is 6.

Thành Đạt Lê by Thành Đạt Lê , 17, Singapore

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2 solutions

There are two possibilities. [1] AB||DC, or [2] BC||AD.
Try [1]:- From A drop a perpendicular AE, E on DC. From B drop a perpendicular BF, F on DC.
In rectangle AEFB, EF=AB=3.
So DE+FC=9. Let DE=X, and we get FC=9-x.
Since height, h=AE=BF,
In right triangles ADE and BFC, 5 2 − X 2 = h = 1 0 2 − ( 9 − X ) 2 . S o X = 1 / 3. h 2 = 5 2 − ( 1 / 3 ) 2 = 224 / 9 I n r i g h t t r i a n g l e B D F , B D 2 = h 2 + D F 2 = 224 / 9 + ( 10 / 3 ) 2 = 36. ∴ B D = 6. 5^2-X^2=h=10^2-(9-X)^2.\\ So~X=1/3.~~~~~~~~~h^2=5^2-(1/3)^2=224/9\\ In ~right~ triangle~ BDF,~~~ BD^2= h^2+DF^2=224/9+(10/3)^2=36.\\ \therefore~~~BD=\color{#D61F06}{6}.


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Ahmad Saad
Aug 24, 2017

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