Another Brick On The Wall...

Geometry Level 2

Two walls on a city collapsed. The only solution found was attaching two ropes on each wall to hold each other, as in the image above. Knowing that the walls have 9 and 3 meters, find the height of the point which the two ropes cross.

1.75 1.50 2.50 2.25

Thiago Virgínio by Thiago Virgínio , 21, Brazil

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1 solution

Tom Engelsman
May 8, 2021

Let us divide the base into y y and z z such that we have two sets of similar right triangles with respect to each wall:

y x = y + z 3 \large \frac{y}{x} = \frac{y+z}{3} (i) and z x = y + z 9 ( i i ) ⇒ 3 y = x ( y + z ) = 9 z ⇒ y = 3 z \large \frac{z}{x} = \frac{y+z}{9} (ii) \Rightarrow 3y = x(y+z) = 9z \Rightarrow y = 3z (iii)

If we now substitute (iii) into either (i) or (ii), we now obtain 3 z x = z + 3 z 3 ⇒ 3 x = 4 3 ⇒ x = 9 4 . \large \frac{3z}{x} = \frac{z + 3z}{3} \Rightarrow \frac{3}{x} = \frac{4}{3} \Rightarrow \boxed{x = \frac{9}{4}}.

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