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Is it possible to find the coordinates of the point marked (?,?) if I have a rectangle/right triangle with a given point and the length of the hypotenuse?

See image:enter image description here

Thanks everyone.

N. F. Taussig
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Michael Seltenreich
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3 Answers3

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Let the point be $(x,y)$. The two lines perpendicular to each other must have the product of their slopes equal to $-1$. Therefore you get $$\frac{y_1-y}{x_1-x}\frac{y_2-y}{x_2-x}=-1$$ Now using distance formula, you can get your second equation $$(x_1-x)^2+(y_1-y)^2+(x_2-x)^2+(y_2-y)^2=h^2$$ When you solve for x,y , you shall get a set of points(locus) satisfying it. In this case, it's a circle. Therefore, the point can be determined but not uniquely.

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The blue rectangle has all the same properties as the rectangle you are looking for: same corners $(x_1, y_1)$ and $(x_2, y_2)$ and same hypotenuse, $h$; but clearly it is not the same rectangle. So no, the given information is not enough to determine the coordinates of the point $(?,?)$, which could be either at the corner of the black rectangle, the corner of the blue rectangle, or at the corner of any of many other rectangles that could be drawn with the same diagonal.

enter image description here

David K
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  • I can't find a flaw in your argument. But could you tell me where did I go wrong? –  Feb 08 '15 at 03:10
  • @GarvilSinghal I see you have figured it out. Well done! – David K Feb 08 '15 at 03:16
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    In fact the arbitrary point may lie anywhere along the locus of the circumcircle that can be constructed using the midpoint of the line segment as the centre and the length of the segment as the diameter. – Deepak Feb 08 '15 at 03:17
  • and what if I can tell the angle between the line h, and the line between (x2,y2) and (?,?). Then could I find the the coordinates? @DavidK – Michael Seltenreich Feb 08 '15 at 03:22
  • @MichaelSeltenreich Yes, if you have that angle you can find the slope of the edge from $(x_1,y_1)$ to $(?,?)$, then use Garvil's first equation to find the slope from $(x_2,y_2)$ to $(?,?)$; a point and a slope gives you an equation of a line; the equations of two lines can be solved simultaneously to find their intersection, which is your answer. – David K Feb 08 '15 at 03:28
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Hammer two nails into the top of your desk or table at the two given points $P_1 = (x1, y1)$ and $P_2 = (x2, y2)$. Place a large rectangular piece of cardboard so that two adjacent sides touch the two nails. The corner where these two sides meet is the point $P= (?,?)$.

enter image description here

Can you slide the cardboard around (causing the point $P$ to move), or is its position fixed?

For extra credit:

  • Mark the midpoint $M$ between $P1$ and $P2$.

  • Mark several points $A$, $B$, $C$, $D$, etc. that $P$ travels to as you move the cardboard.

  • Measure the distance from each of $A$, $B$, $C$, $D$ to $M$.

  • Think about what kind of curve the point $P$ is traveling along as it moves.

bubba
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  • This doesn't seem to answer the question because in the question, the points P1 and P2 need to be at the edges of the yellow square. – Guntram Blohm Feb 08 '15 at 08:51
  • The question is about a rectangle, not about a square. No matter how the cardboard is oriented, the line P1P2 forms the diagonal of a rectangle that has its corner at P. – bubba Feb 08 '15 at 11:02