Properties of Triangles Section 5-3: Bisectors in Triangles

Learning objectives Learn about angle bisectors in triangles, their properties and how to use them to solve problems.

Review: Midesgments; perpendicular bisectors and angle bisectors. Midsegments join two midpoints and are parallel to the third side and half the length of the third side. Perpendicular bisectors connect a midpoint with the opposite vertex and bisect both the line segment and the vertex. Every point on the perpendicular bisector is equidistant from the endpoints of the line segment. Angle bisectors cut an angle in half and are equidistant from the sides of the angle. Draw pictures, reminding them about how similar they are to constructions

Discovery learning! Draw a circle centered at point C, then pick three points on the circle and connect them (making a triangle). Draw three lines from the center of the circle perpendicular to each side of the triangle. What conjectures (educated guesses) can you make about the how the sides of the triangle are divided by those lines? The perpendicular lines divide the sides evenly, making them perpendicular bisectors.

Concurrency in triangles When three or more lines intersect at the same point, they are concurrent. The point at which they intersect is called the point of concurrency. In this case that point is C. For any triangle, certain sets of lines are always concurrent. Two of these sets of lines are the perpendicular bisectors of the triangle’s three sides and the bisectors of the triangle’s three angles. Distance from endpoints to the center are equidistant. Every point on the vertical line from the center of the top of the board to the vertex is as well.

Theorem 5-6: Concurrency of Perpendicular Bisectors Theorem We can use the distance formula to confirm this! The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. The circle is thus circumscribed about the triangle. The circumcenter is just one of many triangle “centers”.

The circumcenter of a triangle an be inside, on or outside the triangle. Acute = inside; right = on; obtuse = outside

What are the coordinates of the circumcenter of the triangle with vertices P(0, 6), O(0, 0), and S(4, 0)? Hint: it’s a right triangle, so where should the center be? Find the intersection point of two of the triangle’s perpendicular bisectors. Here, it is easiest to find the perpendicular bisectors of PO and OS. Because it’s a right triangle it is on one of the sides. Acute would be inside and obtuse would be outside.

If I want to place a bench equidistant from three trees in a park, where should I put it? At the point of concurrency of the perpendicular bisectors of the triangle formed by the trees A, B and C. This point is the circumcenter of triangle ABC and thus equidistant from A, B and C.

Theorem 5-7: Concurrency of Angle Bisectors Theorem We can use the distance formula to confirm this! The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. For any triangle, the incenter is always inside the triangle. In the diagram, points X, Y, and Z are equidistant from P, the incenter of ABC. P is the center of the circle that is inscribed inside the triangle. How many circles can we inscribe inside a triangle that touch all three sides? Just one, where the radius is equal to the distance from the center to the sides.

GE = 2x - 7 and GF = x + 4. What is GD? G is the incenter of ABC because it is the point of concurrency of the angle bisectors. By the Concurrency of Angle Bisectors Theorem, the distances from the incenter to the three sides of the triangle are equal, so GE = GF = GD. Use this relationship to find x.