Center of triangle worksheets for practice finding the centroid of a triangle, orthocenter of a triangle, and circumcenter of a triangle. Checkout the worksheets below, or scroll down for detailed steps on the calculating the various types of triangle centers...
The Centers of Triangles worksheets on this page start with requiring students to check what they have learned about the basic concepts of triangle centers by filling in the blanks on each sentence. A word bank is provided at the bottom of the worksheets. Here’s a tip for you when a word bank is provided: Always do the easiest and most obvious questions first then use the process of elimination.
The second set of worksheets require students to identify the point of concurrency through multiple choice questions. The type of triangle by side and by angle will also be covered on this set.
And finally… worksheets that focus on finding the coordinates of centroid, orthocenter, circumcenter, and incenter are also available. All with answer keys!
Try some of the center of triangles worksheets below, or scroll down for more tips and ways on how to find the center of a triangle.
Refer to the figure above.
The lines from each vertex (corner) to the midpoint of the opposite side of the triangle are the medians. The point where all three lines intersect is the centroid.
Therefore, to find the centroid of a triangle, simply draw the medians (these are line segments that connect any vertex of a triangle to the opposite side). The point where all these three medians of the triangle meet is the centroid.
It is always inside the triangle.
It always divides each median into segments in the ratio of 2:1, with the longer one near the vertex (as shown in the figure above).
It is the point of intersection or concurrency of the three medians of the triangle.
This diagrams illustrates the centroid of a triangle formula:
Where,
Refer to the figure above.
The lines from each vertex (corner) of the triangle and is perpendicular to the opposite side are the triangle’s height or altitude.
The point where all three lines intersect is the orthocenter.
Therefore, to find the orthocenter of a triangle, simply draw the altitudes (these are perpendicular or straight lines that make an angle of 90°) from a triangle's side to the opposite vertex. The point where all these three altitudes of the triangle meet is the orthocenter.
Step 1: Find the equation of one line.
Where,
Step 2: Find the slope of one side.
Where,
Step 3: Repeat Steps 1 and 2 to get the equation of another line.
Step 4: Calculate the slope that is perpendicular to the sides on previous steps
Where,
Step 5: Use the new slopes and the coordinates of the opposite vertices to find the equations of lines that form two altitudes.
Where,
Step 6: Use substitution to solve the system of two equations.
This figure illustrates the circumcenter of a triangle:
The lines from each of the triangle’s midpoint to the opposite side are the triangle’s perpendicular bisectors. The circle that passes through all the vertices of the triangle is the circumcircle.
The center of the circumcircle and where all three lines intersect is the circumcenter.
To find the circumcenter of a triangle, simply draw the perpendicular bisectors (these are line segments which divide a triangle’s side into two equal parts at 90°) from each of triangle’s midpoints to the opposite side. It is not necessary that they should pass through the vertex of a triangle but passes through the midpoint of the sides. The point where all these three perpendicular bisectors of the triangle meet is the circumcenter.
You can use the formula shown in the diagram below to calculate the circumcenter of a triangle:
Where,
This figure illustrates the incenter of a triangle:
The lines from each of the triangle’s vertex to the opposite side are the triangle’s angle bisectors. The biggest circle that can fit inside the triangle is the incircle. The center of the incircle and where all three lines intersect is the incenter.
To find the incenter of a triangle, simply draw the angle bisectors (these are line segments which divide an angle into two equal parts) from each of triangle’s vertices to the opposite side. The point where all these three angle bisectors of the triangle meet is the incenter.
It is always inside the triangle.
It is always equidistant from the sides of the triangle.
The incenter of a triangle is also the center of its incircle.
You can use the formula below to calculate the incenter of a triangle formula:
Where,