Thread: Circumcenter and Orthocenter of a triangle

1. Circumcenter and Orthocenter of a triangle

I searched through the forum and found some threads about this, but it seems like I'm doing everything right but I'm getting an unreasonable answer that doesnt seem right when i match it up with the coordinate i drew before hand

The problem: Triangle ABC with X(73,33) Y(33,35), and Z(52,27), find the circumcenter and Orthocenter of the triangle.

Here is what i did for circumcenter. I found the slope of XY which is -2/40 so the perp slope is 20. The slope of XZ is 6/21 so the perp slope is -21/6.

Then i found the midpt of XY and I got (53,34) and named it as point A. Next, the midpt of XZ is (62.5,30) and I named it B.

And then i set up these two formulas: y = 20x - 1026 (i found the y-intercpt using point A), and y = -21/6x + 248.75 (using pt E).

Then I set up a system: x-4 = 5x+24 and got x=54.245 and substituted it for the first formula and got y=58.9. So i got (54.245,58.9) as the answer.

I tried to match it up but the answer i got, the point was way off, what am i doing wrong? is it the calculations cuz i checked it like 3 times?

Here is what i did for orthocenter. I found the slope of XY which is -2/40 so the perp slope is 20. The slope of XZ is 6/21 so the perp slope is -21/6

Then i used the formulas: y= 20x-1013 (the XY altitude that contains the point Z) and y=-21/6x+150.5(the XZ altitude that contains the point Y).

Then I set up a system: 20x-1013 = -21/6x+150.5 and got x= 49.511 and substituted it for the first formula and got y=22.78. So i got (49.511,22.78) as the answer.

and also like the circumcenter I tried to match it up with the point i contructed using a compass before hand but the point is still off?? why? what am i doing wrong?

any help is appretiated. i am so confused

2. Originally Posted by fuyu1993
I searched through the forum and found some threads about this, but it seems like I'm doing everything right but I'm getting an unreasonable answer that doesnt seem right when i match it up with the coordinate i drew before hand

The problem: Triangle ABC with X(73,33) Y(33,35), and Z(52,27), find the circumcenter and Orthocenter of the triangle.

Here is what i did for circumcenter. I found the slope of XY which is -2/40 so the perp slope is 20. The slope of XZ is 6/21 so the perp slope is -21/6.

Then i found the midpt of XY and I got (53,34) and named it as point A. Next, the midpt of XZ is (62.5,30) and I named it B.

And then i set up these two formulas: y = 20x - 1026 (i found the y-intercpt using point A), and y = -21/6x + 248.75 (using pt E).

Then I set up a system: x-4 = 5x+24 and got x=54.245 and substituted it for the first formula and got y=58.9. So i got (54.245,58.9) as the answer.

I tried to match it up but the answer i got, the point was way off, what am i doing wrong? is it the calculations cuz i checked it like 3 times?
What do you mean by "the point was way off"? What are you trying to match it up with? It looks close enough to what I get by construction.

CB

3. Originally Posted by fuyu1993

Here is what i did for orthocenter. I found the slope of XY which is -2/40 so the perp slope is 20. The slope of XZ is 6/21 so the perp slope is -21/6

Then i used the formulas: y= 20x-1013 (the XY altitude that contains the point Z) and y=-21/6x+150.5(the XZ altitude that contains the point Y).

Then I set up a system: 20x-1013 = -21/6x+150.5 and got x= 49.511 and substituted it for the first formula and got y=22.78. So i got (49.511,22.78) as the answer.

and also like the circumcenter I tried to match it up with the point i contructed using a compass before hand but the point is still off?? why? what am i doing wrong?

any help is appretiated. i am so confused
Your solution does not satisfy your equations, try solving them again more carefully.

CB

4. Originally Posted by CaptainBlack
Your solution does not satisfy your equations, try solving them again more carefully.

CB
do u mean my calculations are wrong?

5. Originally Posted by CaptainBlack
What do you mean by "the point was way off"? What are you trying to match it up with? It looks close enough to what I get by construction.

CB
really? cuz the point i constructed lay just right outside of XY and clearly the point that i tired to calculated doesnt match

6. Originally Posted by fuyu1993
do u mean my calculations are wrong?
Why are you asking me, have you checked that the calculated point satisfies the equations?

CB

7. Originally Posted by fuyu1993
really? cuz the point i constructed lay just right outside of XY and clearly the point that i tired to calculated doesnt match
That sentence does not make sense, and yes really.

CB

8. Originally Posted by CaptainBlack
Why are you asking me, have you checked that the calculated point satisfies the equations?

CB
yes it actually does but idk it still doesnt match the dot.
the coordinate that i constructed was (51,24)

9. Originally Posted by CaptainBlack
That sentence does not make sense, and yes really.

CB
lol idk how to explain it sorry =[
and the coordinate i constructed is (54,35) with some decimals

10. Hi, fuyu1993,

Check your calculations for verifying it.

If it's the centre of a circle, then it has to be the same distance from
all 3 given points.
You can use Pythagoras' theorem for that.
Alternatively, you may verify that the point you found lies on all 3 perpendicular bisectors by placing the x and y you worked out into all 3 equations for the bisectors and verifying that the equations are true for those co-ordinates.

You have made an error calculating the orthocentre.
You have miscalculated when solving the equations of the perpendicular lines that passes through the opposite vertices.

11. Originally Posted by Archie Meade
Hi, fuyu1993,

Check your calculations for verifying it.

If it's the centre of a circle, then it has to be the same distance from
all 3 given points.
You can use Pythagoras' theorem for that.
Alternatively, you may verify that the point you found lies on all 3 perpendicular bisectors by placing the x and y you worked out into all 3 equations for the bisectors and verifying that the equations are true for those co-ordinates.

You have made an error calculating the orthocentre.
You have miscalculated when writing one of the equations of the perpendicular line that passes through the opposite vertex.

for the circumcenter, when i use a compass to constuct, i got the coordinate (54,35) with some decimals and it is the same distance to all 3 points but my calculations seems not to match up with my constuction. why? =[

i cant find the error for orthocentre, i checked like 5 times and eveytime it came out with the same answer that dont match my construction

12. The majority of the circle lies above the triangle.
if you sketch it, the y co-ordinate of the centre is significantly above the triangle. Your check is in error somehow.

13. Here's a rough sketch of the circumcentre.

14. Originally Posted by Archie Meade
The majority of the circle lies above the triangle.
if you sketch it, the y co-ordinate of the centre is significantly above the triangle. Your check is in error somehow.
wait, are u talking about the orthocentre?
i still dont get it

15. That's a picture of the circumcentre.
To find that, use your compass to bisect any 2 sides.
Draw the 2 perpendicular bisectors and locate the point of intersection.

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