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How to determine point is Inside the Triangle?

How do you find the circumcenter of a triangle WITHOUT drawing or graphing it?

  • What steps do you need to take? I know that the circumcenter is the point where the right bisectors intersect, and you first have to find the midpoints of the lines, then the equation of the right bisectors, but how do you find out where the point is exactly, without drawing or graphing it? For example, with these coordinates: A(5,1) B(-2,0) and C(4,8) where ABC are the verticies of the triangle How would you then determine the coordinates of the circumcenter of triangle ABC?

  • Answer:

    You can do it by figuring out the equation of two perpendicular lines that are the bisectors of a couple sides, say AB and BC. Then figure out where the two lines intersect. It's rather complicated but I can explain all the steps if you like... FINDING THE PERPENDICULAR BISECTOR OF AB: Step 1: Find the midpoint of a side. This is just the average of the x and y coordinates. A(5,1) B(-2,0) midpoint x = (5 + -2)/2 = 3/2 midpoint y = (1 + 0)/2 = 1/2 So the midpoint of AB is (3/2, 1/2) Step 2: Find the slope of that line segment. The slope of AB is the difference of y coords over the difference of the x coords: delta y = y2 - y1 = (0 - 1) = -1 delta x = x2 - x1 = (-2 - 5) = -7 m = slope = -1/-7 = 1/7 Okay, the slope of AB is 1/7, so the perpendicular line will have a slope equal to -7 (the negative reciprocal). Step 3: Find an equation for the perpendicular bisector: Finally, a point (3/2, 1/2) and a slope (-7), so use the point-slope form for this perpendicular bisector: y - y1 = m(x - x1) y - 1/2 = -7(x - 3/2) I like to solve for y at this point to make it cleaner and get it into slope-intercept form: y = -7x + 21/2 + 1/2 y = -7x + 11 Good, one line down, now repeat for another bisector. PERPENDICULAR BISECTOR OF BC: I'll abbreviate the steps a little: Step 1: Midpoint of BC midpoint x = (-2 + 4)/2 = 1 midpoint y = (0 + 8)/2 = 4 midpoint (1,4) Step 2: Slope BC: delta y = 8 delta x = 6 m = 4/3 Step 3: Equation of the bisector: Slope = -3/4 Point = (1, 4) y - 4 = -3/4(x - 1) y = -3/4x + 3/4 + 4 y = -3/4x + 4 3/4 NOW EQUATE THE TWO: y = -7x + 11 y = -3/4x + 4 3/4 -7x + 11 = -3/4x + 4 3/4 Multiply by 4 to get rid of the fractions: -28x + 44 = -3x + 19 25x = 44 -19 25x = 25 x = 1 y = -7x + 11 y = -7(1) +11 y = 4 So the circumcenter would be at (1,4) I think I have the math all correct now. And now that I have, it is apparent that we have a right triangle.... if I just checked the slope of AC (-7) and compared it to AB (1/7) I would have seen that we have a right triangle. And in that case, the circumcenter will be equivalent to the midpoint of the other side BC. Hmm... Lo and behold, that was (1,4)... Oh well, now you have the general solution and the specific solution.

amanda_b... at Yahoo! Answers Visit the source

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Follow through mathematically what you state. 1. Find the equation of the perpendicular bisector of the side (5,1) to (-2,0) You already have 2 pieces of info: a) slope is 90 degrees from the slope of AB. slope AB = (1-0)/(5+2)= 1/7. hence slope of right bisector is -7. b) this line goes through the midpoint of AB which is ((5-2)/2, (1+0)/2), which is (3/2,1/2) With these two pieces, find the equation of the line. b) Repeat for side BC and get another equation linking y and x c. Where the lines intersect, the two equations are both valid. So solve the two equations as simultaneous equations and you have the answer. This technique is a good starter because it reflects what you've articulated in words. Once you're confident, search for other techniques.

astatine

You already answered your own question. Drawing or graphing only helps you do it quickly without a mistake while keeping the concept clear in your head because you happened to be looking at it. Find the equation of the lines between the three points and then the midpoints. Find the Perpendicular bisectors (which will obviously go through the midpoints) and then see where the intersect. This can all be done algebraically without drawing a single picture.

The Prince

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