How to find a radius with cordinates?

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Ana.stasia

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I have the cordinates of the vertices of the triangle. How do I find the R?
IMG_20210202_212612.jpg
 
Ana.stasia said:
I have the coordinates of the vertices of the triangle. How do I find the R?
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That picture is definitely not to scale!

I don't know of any quicker method than to find the circumcenter geometrically: the intersection of the perpendicular bisectors of two of the lines AB, BC, and AC.

There are formulas for the circumradius using side lengths and/or angles, e.g. here: https://en.wikipedia.org/wiki/Circumscribed_circle#Other_properties. But I am not aware of any direct way to find it from the vertices. In this case, I find that the circumcenter is at nice coordinates, so I imagine you are expected to find that first.
 
These three formulas are all I have been given. Is there any way to use them to solve this?
IMG_20210202_225632.jpg
 
Ana.stasia said:
These three formulas are all I have been given. Is there any way to use them to solve this?
View attachment 24839
Click to expand...
Formulas, unfortunately, mean nothing if their parts are not defined. I hope you were taught what these formulas mean!

The first formula is the distance formula, giving the distance d between points \((x_1,y_1)\) and \((x_2, y_2)\). That will be useful to find the length of the radius.

The second set of formulas identify a point \((x,y)\) that divides the line segment from \((x_1,y_1)\) to \((x_2, y_2)\) in the proportion \(\lambda:1\). The special case where \(\lambda=1\) is the midpoint formula, which is useful for finding the perpendicular bisector.

The third formula gives the area of the triangle with the given coordinates. You could use it if you also knew the second formula in the link I provided, which relates circumradius R to the area and the side lengths a, b, c. But that isn't a formula I myself know without looking it up.

I suggest using the method I showed, which involves finding the equations of two perpendicular bisectors, as you appear to have the required knowledge for that.
 
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