Notes
Midsegments
A midsegment joins two side midpoints. It is parallel to the third side and half its length.
Special segments and centers
| Segments | Meet at | Key fact |
|---|---|---|
| Perpendicular bisectors | Circumcenter | Equidistant from the vertices |
| Angle bisectors | Incenter | Equidistant from the sides |
| Medians | Centroid | $\displaystyle \frac{2}{3}$ of each median from the vertex |
| Altitudes | Orthocenter | Lines through vertices, perpendicular to opposite sides |
- Points on a perpendicular bisector are equidistant from the segment's endpoints. Points on an angle bisector are equidistant from the angle's sides.
- Centroid on a grid: average the three vertices.
- The circumcenter of a right triangle is the midpoint of the hypotenuse.
Inequalities
- The largest angle is opposite the longest side.
- Triangle Inequality: any two sides add to more than the third. Third side: $|a - b| < c < a + b$.
- Hinge Theorem: with two pairs of congruent sides, the larger included angle gives the longer third side.
- Indirect proof: assume the opposite and reach a contradiction.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Each midsegment is half a side, so the perimeter is half of $48$.
$CG = 2 \cdot GF$: $4x = 2x + 6$, so $x = 3$. $CG = 12$ and $GF = 6$, so $CF = 18$.
$11 - 6 < x < 11 + 6$, with strict inequalities.
The smallest angle is opposite the shortest side, $\overline{XY}$, which is $\angle Z$.
The incenter (and centroid) are always inside. The circumcenter and orthocenter can be outside an obtuse triangle.
$3x + 1 = 5x - 7$ gives $x = 4$, so the distance is 13.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Three friends live at points on a map grid (units in miles): $A(0, 0)$, $B(8, 0)$ and $C(0, 6)$. They form a right triangle with the right angle at $A$.
(a) They want to meet at a spot the same distance from all three homes. Name the point of concurrency and find its coordinates.
Show the answer and scoring guide
The circumcenter. In a right triangle it is the midpoint of the hypotenuse $\overline{BC}$: $\displaystyle \left(\frac{8 + 0}{2}, \frac{0 + 6}{2}\right) = (4, 3)$.
Check off the points you earned:
(b) How far does each friend travel to that spot in a straight line?
Show the answer and scoring guide
Half the hypotenuse: $\displaystyle BC = \sqrt{8^2 + 6^2} = 10$, so each travels 5 miles. Check from $A$: $\displaystyle \sqrt{4^2 + 3^2} = 5$.
Check off the points you earned:
(c) Find the centroid of the triangle. Is it the same point as in part (a)?
Show the answer and scoring guide
$\displaystyle \left(\frac{0 + 8 + 0}{3}, \frac{0 + 0 + 6}{3}\right) = \left(\frac{8}{3}, 2\right)$. No, it is a different point from $(4, 3)$.
Check off the points you earned: