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Difference between revisions of "1960 IMO Problems/Problem 7"

(New page: ==Problem== An isosceles trapezoid with bases <math>a</math> and <math>c</math> and altitude <math>h</math> is given. a) On the axis of symmetry of this trapezoid, find all points <math>P...)
 
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a) On the axis of symmetry of this trapezoid, find all points <math>P</math> such that both legs of the trapezoid subtend right angles at <math>P</math>;
 
a) On the axis of symmetry of this trapezoid, find all points <math>P</math> such that both legs of the trapezoid subtend right angles at <math>P</math>;
  
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b) Calculate the distance of <math>p</math> from either base;
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b) Calculate the distance of <math>P</math> from either base;
  
 
c) Determine under what conditions such points <math>P</math> actually exist. Discuss various cases that might arise.
 
c) Determine under what conditions such points <math>P</math> actually exist. Discuss various cases that might arise.
  
 
==Solution==
 
==Solution==
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{{solution}}
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(a) The intersection of the circle with diameter one of the legs with the axis of symmetry.
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(b) Let <math>x</math> be the distance from <math>P</math> to one of the bases; then <math>h - x</math> must be the distance from <math>P</math> to the other base. Similar triangles give <math>\frac{x}{\frac{a}{2}} = \frac{\frac{c}{2}}{h - x}</math>, so <math>x^2 - hx + \frac{ac}{4} = 0</math> and so <math>x = \frac{h \pm \sqrt{h^2 - ac}}{2}.</math>
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(c) When <math>h^2 \ge ac</math>.
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<center>
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[https://www.artofproblemsolving.com/Wiki/images/0/01/1960_7.jpg]
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</center>
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In our above picture, ABCD is our trapezoid with AB=a and CD=c and its height is 'h'.  AF and BE are perpendicular to CD such that AF= BE= h.  XY is our axis of symmetry and it intersects with CD at a point O. Point O is our origin of reference whose coordinates are (0,0).
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Let our point P be on the axis of symmetry at z distance from the origin O.
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The coordinates of the points A,B,C,D,E,F and P are given in the figure.
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Now,
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Slope of the line <math>PC= (z-0)/(0-c/2) =  -2z/c</math>
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Slope of the line <math>PB= (z-h)/(0-a/2) = -2(z-h)/a</math>
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Since the leg BC subtends a right angle at P, the angle BPC should be a right angle. This means that the product of the slope of PC and PB is -1.
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i.e
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<math>4z(z-h)=-ac</math>
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or <math>z^2 - zh + ac/4= 0</math>
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Now, solving for z, we get,  <math>z=  [(h + ( h^2 - ac ) ^1/2 ]/2</math>  and  <math>[(h - ( h^2 - ac ) ^1/2 ]/2</math>
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So, z is the distance of the points from the base  CD..
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Also the points are possible only when , <math>h^2 - ac >= 0</math>.. and doesn't exist for <math>h^2 -ac <0</math>
  
 
==See Also==
 
==See Also==
  
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{{IMO box|year=1960|num-b=6|after=Last Question}}
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{{IMO7 box|year=1960|num-b=6|after=Last Question}}
  
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]

Latest revision as of 14:25, 23 March 2020

Problem

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.

a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;

b) Calculate the distance of $P$ from either base;

c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.

Solution

(a) The intersection of the circle with diameter one of the legs with the axis of symmetry.

(b) Let $x$ be the distance from $P$ to one of the bases; then $h - x$ must be the distance from $P$ to the other base. Similar triangles give $\frac{x}{\frac{a}{2}} = \frac{\frac{c}{2}}{h - x}$, so $x^2 - hx + \frac{ac}{4} = 0$ and so $x = \frac{h \pm \sqrt{h^2 - ac}}{2}.$

(c) When $h^2 \ge ac$.

[1]

In our above picture, ABCD is our trapezoid with AB=a and CD=c and its height is 'h'. AF and BE are perpendicular to CD such that AF= BE= h. XY is our axis of symmetry and it intersects with CD at a point O. Point O is our origin of reference whose coordinates are (0,0).

Let our point P be on the axis of symmetry at z distance from the origin O.

The coordinates of the points A,B,C,D,E,F and P are given in the figure.

Now,

Slope of the line $PC= (z-0)/(0-c/2) = -2z/c$ Slope of the line $PB= (z-h)/(0-a/2) = -2(z-h)/a$

Since the leg BC subtends a right angle at P, the angle BPC should be a right angle. This means that the product of the slope of PC and PB is -1.

i.e

$4z(z-h)=-ac$

or $z^2 - zh + ac/4= 0$

Now, solving for z, we get, $z= [(h + ( h^2 - ac ) ^1/2 ]/2$ and $[(h - ( h^2 - ac ) ^1/2 ]/2$

So, z is the distance of the points from the base CD..

Also the points are possible only when , $h^2 - ac >= 0$.. and doesn't exist for $h^2 -ac <0$

See Also

1960 IMO (Problems)
Preceded by
Problem 6 		1 • 2 • 3 • 4 • 5 • 6 • 7 Followed by
Last Question
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