Unit 5: Triangle Relationships and Centers
Four special points hiding inside every triangle, built from bisectors, medians, and altitudes, and the surprising line that connects three of them.
7 topics · 25 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Perpendicular Bisectors and the Circumcenter
- How it worksHow do the three perpendicular bisectors of a triangle's sides all cross at one single point?
- Worked exampleFind the circumcenter of a triangle with vertices at (0, 0), (6, 0), and (0, 8).
- Common mistakeWhy isn't the perpendicular bisector of a triangle's side the same line as the angle bisector from the opposite vertex?
- Going furtherWhat happens to the circumcenter's location when a triangle changes from acute to obtuse?
Your turn1 of 2
A right triangle has vertices at (0, 0), (15, 0), and (0, 14). Find the sum of the coordinates of its circumcenter.Angle Bisectors and the Incenter
- How it worksHow do the three angle bisectors of a triangle always meet at a single interior point?
- Worked exampleFind the radius of the inscribed circle of a right triangle with legs 6 and 8 and hypotenuse 10.
- Worked exampleFind the distance from the incenter to each side of a triangle whose inscribed circle has radius 4.
- Common mistakeWhy doesn't the incenter have to be the same point as the triangle's centroid?
Your turn1 of 2
A right triangle has legs of length 15 and 14. Find the radius of its inscribed circle.Medians and the Centroid
- How it worksHow does each median connect a vertex to the midpoint of the opposite side?
- Worked exampleFind the centroid of a triangle with vertices at (0, 0), (6, 0), and (3, 9).
- Common mistakeWhy doesn't the centroid sit halfway along a median instead of two-thirds of the way from the vertex?
- Going furtherWhat happens to the two segments of a median when the centroid divides it?
Your turn1 of 2
A triangle has vertices at (0, 0), (15, 0), and (3, 19). Find the sum of the coordinates of its centroid.Altitudes and the Orthocenter
- How it worksHow does an altitude drop a perpendicular from one vertex all the way to the line containing the opposite side?
- Worked exampleFind the orthocenter of a right triangle with the right angle at vertex B.
- Common mistakeWhy doesn't an altitude of a triangle have to hit the opposite side directly instead of its extension?
- Going furtherWhat happens to the orthocenter's position as a triangle stretches from acute to obtuse?
Your turn1 of 2
A right triangle has its right angle at vertex B, located at (4, 3). Find the sum of the coordinates of the triangle's orthocenter.The Euler Line
- How it worksHow do the centroid, circumcenter, and orthocenter stay collinear no matter how the triangle is reshaped?
- Worked exampleFind the centroid's position on the Euler line given a circumcenter at (0, 0) and an orthocenter at (9, 6).
- Common mistakeWhy isn't the incenter also guaranteed to fall on the Euler line with the other three centers?
Your turn1 of 2
A triangle's circumcenter is at (0, 0) and its orthocenter is at (22, 21). Find the sum of the coordinates of the centroid, which lies on the Euler line one-third of the way from the circumcenter to the orthocenter.The Triangle Inequality Theorem
- How it worksHow does comparing the sum of any two sides against the third side decide if a triangle can exist?
- Worked exampleFind out whether sides of length 4, 7, and 12 can form a triangle.
- Common mistakeWhy doesn't a side length exactly equal to the sum of the other two ever complete a real triangle?
Your turn1 of 2
Two sides of a triangle measure 15 and 14. Find the smallest whole-number length the third side could have, following the Triangle Inequality Theorem.Midsegments of Triangles
- How it worksHow does a triangle's midsegment end up exactly half the length of the side it's parallel to?
- Worked exampleFind the length of a midsegment parallel to a triangle's side that measures 18.
- Ties togetherHow does the midsegment theorem connect to the same parallel and proportional ideas behind similar triangles?
Your turn1 of 2
A triangle's side measures 28. Find the length of the midsegment parallel to that side.