Honors Geometry Section 4.6 Special Segments in Triangles

Goals for today’s class: 1 Goals for today’s class: 1. Understand what a median, altitude and midsegment of a triangle are. 2. Correctly sketch medians and altitudes in a triangle and identify any congruent segments or angles that result. 3. Write the equation for the line containing a median or altitude given the coordinates of the vertices of the triangle.

*When three or more lines intersect at a single point, the lines are said to be __________ and the point of intersection is called the _________________. concurrent point of concurrency

*A median of a triangle is a segment from a vertex to the midpoint of the opposite side. The medians of a triangle are concurrent at a point called the ________. centroid

*An altitude of a triangle is a segment from a vertex perpendicular to the line containing the opposite side. We have to say “the line containing the opposite side” instead of “the opposite side” because altitudes sometimes fall outside the triangle

Examples: Sketch the 3 altitudes for each triangle. *The point of concurrency for the lines containing the altitudes is called the orthocenter.

While the median and altitude from a particular vertex will normally be different segments, that is not always the case. The median and altitude from the vertex angle of an isosceles triangle will be the same segment.

A midsegment of a triangle is segment joining the midpoints of two sides of a triangle.

Theorem 4. 6. 9. Midsegment Theorem Theorem 4.6.9 Midsegment Theorem A midsegment of a triangle is parallel to the third side and half as long as the third side.

Example: Find the values of all variables:

If two lines are parallel, their slopes are _______ If two lines are parallel, their slopes are _______.   If two lines are perpendicular, their slopes are ___________________   Slope-Intercept form of the equation of a line: __________________   Point-Slope form of the equation of a line: _______________________

a) Find the length of the median from vertex A

b) Write the equation of the line containing the median from vertex A.

c) Write the equation of the line containing the altitude from vertex A.