Section 5 3 Concurrent Lines Medians and Altitudes

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Section 5 – 3 Concurrent Lines, Medians, and Altitudes Objectives: To identify properties of

Section 5 – 3 Concurrent Lines, Medians, and Altitudes Objectives: To identify properties of perpendicular bisectors and angle bisectors To identify properties of medians and altitudes of triangles

Concurrent: When three or more lines intersect in one point. Point of Concurrency: The

Concurrent: When three or more lines intersect in one point. Point of Concurrency: The point at which concurrent lines intersect.

Theorem 5 - 6 The perpendicular bisector of the sides of a triangle are

Theorem 5 - 6 The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices. Circumcenter of the Triangle: The point of concurrency of the perpendicular bisectors of a triangle.

Theorem 5 - 7 The bisectors of the angles of a triangle are concurrent

Theorem 5 - 7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Incenter of the Triangle: The point of concurrency of the angle bisectors of a triangle.

Point Q, R, and S are equidistant from the circumcenter, so the circle is

Point Q, R, and S are equidistant from the circumcenter, so the circle is circumscribed about the triangle. Point X, Y, and Z are equidistant from the incenter, so the circle is inscribed in the triangle.

Example 1 Finding the Circumcenter A) Find the center of the circle that you

Example 1 Finding the Circumcenter A) Find the center of the circle that you can circumscribe about ∆OPS.

B) Find the center of the circle that you can circumscribe about the triangle

B) Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (-8, 0), and (0, 6).

C) Find the center of the circle that circumscribes ∆XYZ.

C) Find the center of the circle that circumscribes ∆XYZ.

Example 2 Connection Real-World A) The Jacksons want to install the largest possible circular

Example 2 Connection Real-World A) The Jacksons want to install the largest possible circular pool in their triangular backyard. Where would the largest possible pool be located?

Textbook Page 259 – 260; #1 – 9 (USE GRAPH PAPER)

Textbook Page 259 – 260; #1 – 9 (USE GRAPH PAPER)

Median of a Triangle: A segment whose endpoints are a vertex and the midpoint

Median of a Triangle: A segment whose endpoints are a vertex and the midpoint of the opposite side

Theorem 5 - 8 The medians of a triangle are concurrent at a point

Theorem 5 - 8 The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. Centroid of the Triangle: The point of concurrency of the medians.

Example 3 Finding Lengths of Medians A) D is the centroid of ∆ABC and

Example 3 Finding Lengths of Medians A) D is the centroid of ∆ABC and DE = 6. Find BD. Then find BE.

B) M is the centroid of ∆WOR, and WM = 16. Find WX.

B) M is the centroid of ∆WOR, and WM = 16. Find WX.

Altitude of a Triangle: The perpendicular segment from a vertex to the line containing

Altitude of a Triangle: The perpendicular segment from a vertex to the line containing the opposite side.

Example 4 Identifying Medians & Altitudes A) B)

Example 4 Identifying Medians & Altitudes A) B)

Theorem 5 - 9 The lines that contain the altitudes of a triangle are

Theorem 5 - 9 The lines that contain the altitudes of a triangle are concurrent. Orthocenter of the Triangle: The point of concurrency of the altitudes.

Textbook Page 260; # 11 – 22

Textbook Page 260; # 11 – 22