Medians and Altitudes of Triangles Concept 37

Median – a segment that connects the vertex of the triangle to the midpoint of the opposite side of the triangle. Vocabulary

Median of a Triangle Medians do have one vertex as an endpoint. Centroid – the point at which medians meet at one point. Where is the centroid located? Always inside the triangle.

Medians of a Triangle Theorem Concept

Example 1 The medians of ABC meet at centroid, point D Example 1 The medians of ABC meet at centroid, point D. Find the indicated values. Find BG Find BD =12 = 8

G is the centroid of ABC, AD = 15, CG = 13, and AD CB . Example 2 G is the centroid of ABC, AD = 15, CG = 13, and AD CB . Find the length of each segment. a. AG = b. GD = c. CD = d. GE = e. GB = f. Find the perimeter of ABC 10 5     6.5     13

3. In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Example 1

4. In ΔABC, CG = 4. Find GE. Example 2

Find the centroid of the given triangle. Find the midpoint D of BC. Graph point D. Example 3

Altitude – a segment from a vertex that is perpendicular to the opposite side or to the line containing the opposite side. Vocabulary

Altitudes of a Triangle Altitudes have one vertex as an endpoint. Orthocenter – the point at which altitudes meet at one point. Where is the orthocenter located? Acute Triangle Inside triangle Right Triangle On the vertex of the right angle Obtuse Triangle Outside, behind the obtuse angle

Altitudes of a Triangle Theorem Concept

Altitude through point I. 6. COORDINATE GEOMETRY The vertices of ΔHIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of ΔHIJ. Orthocenter Altitude through point I.   Opposite reciprocal:   Altitude through point J.   Opposite reciprocal:   Altitude through point H.   Opposite reciprocal:   Example 4

7. COORDINATE GEOMETRY The vertices of ΔABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of ΔABC. A. (1, 0) B. (0, 1) C. (–1, 1) D. (0, 0) Example 4

Perpendicular Bisector Circumcenter Angle Bisector incenter Incenter centroid Median Centroid Altitude Orthocenter orthocenter