Medians and Altitudes Use Medians and Altitudes A

Use Medians and Altitudes A median of a triangle is a segment from a

The point of concurrency of the medians of a triangle is the centroid of

Example: Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and

Use Medians and Altitudes Example 1: Use the centroid of a triangle In Triangle

Use Medians and Altitudes Example 2: Use the centroid of a triangle In Triangle

Use Medians and Altitudes Example 3: Use the centroid of a triangle The vertices

Use Medians and Altitudes An altitude of a triangle is the perpendicular segment from

Use Medians and Altitudes The point at which the lines containing the three altitudes

Use Medians and Altitudes The orthocenter is located in different areas depending on what

Example 3: Finding the Orthocenter Find the orthocenter of ∆XYZ with vertices X(3, –

Example 3 Continued Step 2 Find an equation of the line containing the altitude

Example 3 Continued Step 3 Find an equation of the line containing the altitude

Example 3 Continued Step 4 Solve the system to find the coordinates of the

Altitude. . Vertex. . 90°. . Orthocenter Angle Bisector. . Angle into 2 equal

Give the best name for AB A | B B Median Altitude A A

Vocabulary Review Circumcenter – Point where three perpendicular bisectors of a triangle intersect. Incenter

Slides: 18

Download presentation

Medians and Altitudes

Use Medians and Altitudes A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

The point of concurrency of the medians of a triangle is the centroid of the triangle. The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.

Example: Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find LS. Centroid Thm. Substitute 21 for RL. LS = 14 Simplify.

Example: Using the Centroid to Find Segment Lengths In ∆LMN, RL = 21 and SQ =4. Find NQ. 12 = NQ

Use Medians and Altitudes Example 1: Use the centroid of a triangle In Triangle RST, Q is the centroid and SQ = 8. Find QW and SW.

Use Medians and Altitudes Example 2: Use the centroid of a triangle In Triangle HJK, P is the centroid and JP = 12. Find PT and JT.

Use Medians and Altitudes Example 3: Use the centroid of a triangle The vertices of Triangle FGH are F(2, 5), G(4, 9), and H(6, 1). Which ordered pair gives the coordinates of the centroid P of Triangle FGH?

Use Medians and Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

Use Medians and Altitudes The point at which the lines containing the three altitudes of a triangle intersect is called the orthocenter of the triangle.

Use Medians and Altitudes The orthocenter is located in different areas depending on what type of triangle you have.

Example 3: Finding the Orthocenter Find the orthocenter of ∆XYZ with vertices X(3, – 2), Y(3, 6), and Z(7, 1). Step 1 Graph the triangle. X

Example 3 Continued Step 2 Find an equation of the line containing the altitude from Z to XY. Since XY is vertical, the altitude is horizontal. The line containing it must pass through Z(7, 1) so the equation of the line is y = 1.

Example 3 Continued Step 3 Find an equation of the line containing the altitude from Y to XZ. The slope of a line perpendicular to XZ is pass through Y(3, 6). . This line must Point-slope form. Substitute 6 for y 1, x 1. Distribute . Add 6 to both sides. for m, and 3 for

Example 3 Continued Step 4 Solve the system to find the coordinates of the orthocenter. Substitute 1 for y. Subtract 10 from both sides. 6. 75 = x Multiply both sides by The coordinates of the orthocenter are (6. 75, 1).

Altitude. . Vertex. . 90°. . Orthocenter Angle Bisector. . Angle into 2 equal angles. . Incenter Perpendicular Bisector… 90°. . bisects side. . Circumcenter Median. . Vertex. . Midpoint of side. . Centroid

Give the best name for AB A | B B Median Altitude A A | | B B Angle Perp Bisector

Vocabulary Review Circumcenter – Point where three perpendicular bisectors of a triangle intersect. Incenter – Point where three angle bisectors of a triangle intersect. Centroid (Center of Gravity) – Point where three medians of a triangle intersect. Orthocenter – Point where three altitudes of a triangle intersect.

Related presentations