# Trigonometry basic question

I am training Trigonometry just for fun, so I am not in a hurry, but would like to know how to answer this question - not the result, but how to do it. Sorry because I understand this is too basic for you, but I forgot almost everything from elementary school...

Question says:

A sheet of paper measures $210 \space mm \times 297 \space mm$. Consider a diagonal from one corner of the sheet to its opposite corner, and choose a point on the diagonal so that its distance to the furthest edge of the sheet is equal to the length of that edge, that is, $210 \space mm$. What is the distance of the point to the nearest edge of the sheet?

This is the point I reached (did it in Paint - sorry about the quality):

So as you can see I tried but with no success, and would like to know how to continue. Someone can help me?

Thank you very much in advance.

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$x=\sqrt{297^2+210^2}-210$ –  pedja Dec 5 '11 at 12:35
Wouldn't "distance to the furthest edge" be from the point to one of the edges (not a corner)? So, the vertical line from $x$ to the bottom edge is 210mm. If this is the case, you want to use similar triangles. –  David Mitra Dec 5 '11 at 12:55
To see what David meant, imagine a circle around your point and let the radius grow from 0 until the circle hit the boundary - that radius is the distance you search for. –  AD. Dec 5 '11 at 14:04

Wouldn't "distance to the furthest edge" be from the point to one of the edges (not a corner)? So, we can take the vertical line from your 'x" mark to the bottom edge to have length 210mm. If this is the case, you want to use similar triangles.

Warning: solution follows:

Draw a horizontal line from the point denoted by the "x" mark on your diagram to the right side of your rectangle and a vertical line from the point to the bottom side. This gives you two similar triangles: the "bottom" one, call it $B$, and the "top one", call it $T$.

Let $a$ be the length of the horizontal side of $B$ and let $h$ be the height of the vertical side of $T$. Let $c$ be the length of the horizontal side of $T$ (see diagram below).

As remarked above, we set things up so that the vertical side of $B$ has length $210$ and $h=87$.

A ratio of the non-hypotenuse sides of $B$ is ${210\over a}$.

The corresponding ratio of the non-hypotenuse sides of $T$ is ${87\over c}$.

The corresponding ratio of the non-hypotenuse sides of the lower triangle formed by the diagonal and the rectangle is ${297\over210}$.

Since the three triangles are similar $${210\over a}={87\over c}={297\over 210}.$$ From this, you can find the values of $a$ and $c$. Then you can find the shortest distance to an edge.

Not to scale.

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