Question

3. What are the coordinates of the circumcenter
of AABC?
4. What type of triangle will have a
circumcenter outside the triangle?
5. Where is the incenter of a triangle
always located?

Help please!!!

Answer 1

Answer:

Step-by-step explanation:

3. To find the circumcenter, ie, the center of the circle passing for all 3 corners, let's find the intersection of two right bisectors. (Remember that all the points in the right bisector have the same distance from either ends). The easiest one to use is the bisector of side AB, since it's horizontal: the equation is $x= \frac{x_B+x_A}2 = \frac{3-5}2 = -1$. To find another let's choose a different side, ie AC. We have two ways. Either pick a point in the plane, and impose that the distance from A is the same from the distance from C, or grab the slope of the line AC, the coordinates of the midpoint, and then write down the equation of the perpendicular passing for the midpoint. Let's go for distance. Call $P(x,y)$ the point we want, $\overline {AP} = \overline {CP} \\\sqrt{(x-(-5))^2 +(y-(-4))^2} = \sqrt{(x-(-1))^2+(y-6))^2}\\x^2+10x+25+y^2+8y+16 = x^2+2x+1+y^2-12y+36\\10x+8y+41=2x-12y+35 \\8x+20y+6=0$

Intersecting it with the other bisector $x=-1$ we find$-8+20y+6=0 \rightarrow 20y-2=0 \rightarrow y=\frac1{10}$ giving a circumcenter H coordinates $H(-1; \frac1{10})$

4. As long as the greatest angle in the triangle is more than 90° the circumcenter will be outside the triangle.

5. The incenter, being the intersection of the angular bisector, is always inside the triangle.