Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$?
as discussed in Method III here,
http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Lehman/emat6690/trisecttri's/trisect.html
your point D will be the centroid of ABC. To find the centroid, read here:
http://www.mathopenref.com/coordcentroid.html
Thanks :)
Triangle ABC with vertices A(3,2), B(5,4), C(5, 8) is translated (x-6, y-5). What are the coordinates of triangle A’B’C’?
To find the coordinates of point $D$, we need to find the centroid of triangle $ABC$. The centroid of a triangle is the intersection point of its medians, which are the line segments connecting each vertex to the midpoint of the opposite side.
1. Find the midpoint of each side of triangle $ABC$:
- Midpoint of $AB$: $M_{AB} = \left(\frac{5 + 3}{2}, \frac{8 + (-2)}{2}\right) = (4, 3)$
- Midpoint of $AC$: $M_{AC} = \left(\frac{5 + 6}{2}, \frac{8 + 1}{2}\right) = \left(\frac{11}{2}, \frac{9}{2}\right)$
- Midpoint of $BC$: $M_{BC} = \left(\frac{3 + 6}{2}, \frac{-2 + 1}{2}\right) = \left(\frac{9}{2}, -\frac{1}{2}\right)$
2. Find the coordinates of the centroid using the midpoints:
- Centroid $G = \left(\frac{5 + \frac{11}{2} + \frac{9}{2}}{3}, \frac{8 + \frac{9}{2} - \frac{1}{2}}{3}\right) = \left(\frac{9}{2}, \frac{5}{2}\right)$
Now that we have the coordinates of the centroid $G$, we can find the coordinates of point $D$ inside the triangle. The three small triangles $ABD$, $ACD$, and $BCD$ will have equal areas if the lines $GD$, $AD$, and $BD$ intersect at the same point.
3. Use the centroid $G$ and the equation of the line $AD$ to find the coordinates of $D$:
- Equation of line $AD$: $y = mx + n$
- Substitute the coordinates of $G$ into the equation: $\frac{5}{2} = \frac{9}{2}m + n$
- Rearrange the equation to express $n$ in terms of $m$: $n = \frac{5}{2} - \frac{9}{2}m$
4. Substitute the equation for $n$ into the equation of line $AD$ to get the coordinates of point $D$:
- $y = mx + \left(\frac{5}{2} - \frac{9}{2}m\right)$
- Simplify to find the coordinates of $D$: $D = \left(m, mx + \frac{5}{2} - \frac{9}{2}m\right)$
To find the value of $10m + n$, substitute the coordinates of point $D$ into the expression $10m + n = 10m + \frac{5}{2} - \frac{9}{2}m$ and simplify the expression.
Therefore, the value of $10m + n$ is $\boxed{\frac{10}{2} = 5}$.