Concurrent Lines, Medians, Altitudes Sections 5-3 & 5-4 Concurrent Lines, Medians, Altitudes Objectives: Identify properties of perpendicular and angle bisectors Identify properties of medians and altitudes of triangles Vocabulary Concurrent three or more lines intersect in one point Point of Concurrency the point at which the concurrent lines intersect Point of Concurrency

Concurrency and Perpendicular/Angle Bisectors Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Concurrency and Angle Bisectors Theorem 5-7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

Concurrency and Perpendicular Bisectors The figure shows perpendicular bisectors concurrent at S. The point S is called the circumcenter of the triangle. Points A, B, and C are equidistant from point S. The circle is circumscribed about the triangle.

Concurrency and Angle Bisectors The figure shows angle bisectors concurrent at I. The point I is called the incenter of the triangle. Points A, B, and C are equidistant from point I. The circle is inscribed in the triangle.

Apply Perpendicular Bisectors Find the center of the circle that circumscribes ∆XYZ. Find the perpendicular bisectors (Line XY) y = 4 (Line XZ) y = 3 The lines y = 4 and x = 3 intersect at the point (3, 4). This point is the center of the circle that circumscribes ∆XYZ.

Real-world and Angle Bisectors City planners want to locate a fountain equidistant from three straight roads that enclose a park. Explain how they can find the location. The roads form a triangle around the park. Theorem 5-7 states that the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. The city planners should find the point of concurrency of the angle bisectors of the triangle formed by the three roads and locate the fountain there.

Median of a Triangle 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. The point of concurrency of the medians is called centroid. Point G is the centroid. Medians AG = 2/3 AD CG = 2/3 CF BG = 2/3 BE

Apply Median of a Triangle M is the centroid of ∆WOR, and WM = 16. Find WX. The centroid is the point of concurrency of the medians of a triangle. 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. (Theorem 5-8) Because M is the centroid of WOR, WM = WX. 2 3 WM = WX Theorem 5-8 2 3 16 = WX Substitute 16 for WM. 2 3 24 = WX Multiply each side by . 3 2

Altitude of a Triangle In a triangle, the perpendicular from a vertex to the opposite side is called the Altitude. The altitude can be a side of a triangle or may lie outside the triangle. Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent.

Altitude of a Triangle Theorem 5-9 The lines that contain the altitudes of a triangle are concurrent. The point where the altitudes are concurrent are called the orthocenter of the triangle.

Is KX a median, an altitude, neither, or both? Altitude of a Triangle Is KX a median, an altitude, neither, or both? Because LX = XM, point X is the midpoint of LM, and KX is a median of KLM. Because KX is perpendicular to LM at point X, KX is an altitude. So KX is both a median and an altitude.

Median goes from vertex to midpoint of segment opposite. Compare Medians and Altitudes Median goes from vertex to midpoint of segment opposite. Altitude is a perpendicular segment from vertex to segment opposite.