Medians and Altitudes of Triangles And Inequalities in One Triangle

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Presentation transcript:

Medians and Altitudes of Triangles And Inequalities in One Triangle Chapter 5.2 and 5.3 Medians and Altitudes of Triangles And Inequalities in One Triangle

Definitions Median of a Triangle – a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. Centroid – The point of concurrency of the medians of a triangle.

Concept

In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Use the Centroid Theorem In ΔXYZ, P is the centroid and YV = 12. Find YP and PV. Centroid Theorem YV = 12 Simplify.

In ΔLNP, R is the centroid and LO = 30. Find LR and RO. A. LR = 15; RO = 15 B. LR = 20; RO = 10 C. LR = 17; RO = 13 D. LR = 18; RO = 12

Use the Centroid Theorem In ΔABC, CG = 4. Find GE.

In ΔJLN, JP = 16. Find PM. A. 4 B. 6 C. 16 D. 8

Example 3 Find the Centroid on a Coordinate Plane SCULPTURE Lee is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance? Understand You need to find the centroid of the triangle. This is the point at which the triangle will balance.

Example 3 Find the Centroid on a Coordinate Plane

Example 3 Find the Centroid on a Coordinate Plane The centroid P is the distance. So, the centroid is (2) or units to the right of A. The coordinates are . P

Definition Altitude of a triangle - segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. (aka height)

Concept

Concept

Concept

Concept

Concept

Use the Exterior Angle Inequality Theorem

Use the Exterior Angle Inequality Theorem

A. B. C. D.

A. B. C. D.

Concept

List the angles of ΔABC in order from smallest to largest. Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest. Answer: C, A, B

List the angles of ΔTVX in order from smallest to largest. A. X, T, V B. X, V, T C. V, T, X D. T, V, X

List the sides of ΔABC in order from shortest to longest. Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. Answer: AC, AB, BC

List the sides of ΔRST in order from shortest to longest. A. RS, RT, ST B. RT, RS, ST C. ST, RS, RT D. RS, ST, RT