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Math Challenge II-A Geometry

 
 
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Math Challenge II-A Geometry
by Benjamin Yu - Monday, 17 June 2024, 1:25 AM
 
Can I have some help for question 3.24 and 3.25 please?
 
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Re: Math Challenge II-A Geometry
by Dr. Kevin Wang - Monday, 17 June 2024, 11:00 AM
 

3.24: The question is: Suppose $\triangle ABC$ is an isosceles triangle with $A=(0,2)$ and $B=(2,1)$.  Find all possible coordinates for $C$.

In this question, there are infinitely many possible isosceles triangles using $A$ and $B$ as vertices.  For the location of $C$, there are three cases: (1) $C$ is on the perpendicular bisector of $\overline{AB}$; (2) $C$ is on the circle centered at $A$ with radius $\overline{AB}$; (3) $C$ is on the circle centered at $B$ with radius $\overline{AB}$.  Write the equation for each of these three cases, and the answer is all the possible coordinates of point on these line or circles, except for a few points that makes the triangle degenerate (you need to identify those points too).

3.25: It is easier if you draw a diagram.  From the question, you can determine that $\ell_1$ and $\ell_2$ are parallel to each other, and $\ell_3$ intersects with both of them.  The question asks for the biggest circle that does not go outside $\ell_1$ and $\ell_2$---how do you make that happen?  Then the center of the circle must be on $\ell_3$.  That determines the location of the circle.  Based on those info, you can give the equation of the circle.

Picture of Benjamin Yu
Re: Math Challenge II-A Geometry
by Benjamin Yu - Monday, 17 June 2024, 9:29 PM
 
Thanks.