Finding Coordinates for a Triangle

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Discussion Overview

The discussion revolves around finding the coordinates of the third vertex of an isosceles triangle when two vertices and the height of the triangle are known. Participants explore the implications of different interpretations of the problem, particularly regarding the definitions of "side" and "sides".

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant asks how to find the coordinates of a triangle given two corners and the lengths of the sides, suggesting there may be two solutions.
  • Another participant clarifies the importance of whether "side" or "sides" is referenced, indicating that this distinction affects the number of solutions.
  • A participant states they know the distance between the two points and the height of the triangle, asserting that they can find the other sides since it is an isosceles triangle.
  • Further elaboration is provided on the geometric properties of isosceles triangles, including the relationship between the base, altitude, and midpoint, as well as the method to derive the coordinates of the third vertex.
  • One participant expresses confusion about having two unknowns and one equation, seeking clarification on how to proceed.
  • A response indicates that there is only one equation relating the distance from the midpoint to the height, and suggests using the line equation to solve for the coordinates.

Areas of Agreement / Disagreement

Participants generally agree on the geometric principles involved but express differing interpretations of the problem's parameters, leading to some confusion about the number of solutions and the equations needed to find the coordinates.

Contextual Notes

The discussion highlights potential limitations in the problem's framing, particularly regarding the definitions of "side" and "sides", which may affect the interpretation of the solution space.

powp
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Hello All,

How do you find one set of corrdinates of a trangle when you know two corners and the lengths of the sides?

Is it possible?? It seems that there would be two solutions.

Thanks

P
 
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do you mean "side" or "sides"? since it makes a big difference here.

if you mean "side", and if you mean some characteristic triangle (isoscoles, right, etc.), then yes there will be 2 solutions.

if you mean "sides" then there will clearly be one unique solution of finding the third point.

please clarify you question so that we can help you where you are having a hard time.
 
I know the length between the two point and I know the height of the triange and since it is a isoscoles trianges I can find the other sides.
 
powp said:
I know the length between the two point and I know the height of the triange and since it is a isoscoles trianges I can find the other sides.

In that case, the problem depends strongly upon whether the side you are given is the base.

If you are given the coordinates of the two end points of the base, then, of course, you could calculate the length but you don't really need that. I am going to assume that you are given the coordinates of the two points of the base,(x0,y0) and (x1,y1), and the height of an isosceles triangle.

The altitude of an isosceles triangle passes through the center of the base and is perpendicular to it. Knowing (x0,y0) and (x1,y1), you can find the slope of the line through those two points: [tex]\frac{y_1-y_0}{x_1-x_0)[/tex]. The slope of the line on which the altitude lies is negative reciprocal of that: [tex]\frac{x_0-x_1}{y_1-y_0}[/tex]. Of course, the midpoint of the base is [tex]\(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2}\)[/tex]. The equation of the line through that midpoint having that slope gives you an equation connecting x and y for any point on that line. Use it to make the formula for distance from the midpoint equal to the given height a single quadratic equation for x (or y) and solve.

Yes, there will be two solutions on opposite sides of the base.
 
I am not getting this. Can you please help a bit more. I have two unknown and one equation who do I solve?
 
You don't have two unknowns. The distance from (x,y) to the midpoint equals the given height- that's you one equation. However, you also have the equation of the line (x,y) lies on. That is y= mx+ b for m and b you can calculate. Replace y by mx+b and solve for x. After you find x, you can calculate y= mx+b.
 

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