Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the opposite side.

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Presentation transcript:

Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the opposite side

Just to make sure we are clear about what an opposite side is….. Given ABC, identify the opposite side of A. of B. of C. BC AC AB

Any triangle has three medians. B Any triangle has three medians. L M A N C Let L, M and N be the midpoints of AB, BC and AC respectively. CL, AM and NB are medians of ABC.

Where 3 or more lines intersect A new term… Point of concurrency Where 3 or more lines intersect

Centroid Centroid The point where all 3 medians intersect Is the point of concurrency

The centroid is the center of balance for the triangle. You can balance a triangle on the tip of your pencil if you place the tip on the centroid

Centroid Theorem The centroid of a triangle divides the median into segments with a 2:1 ratio. The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint.

VERTEX 3x 2x CENTROID x MIDPOINT

The distance from the vertex to the centroid is two-thirds Centroid Theorem The distance from the vertex to the centroid is two-thirds the distance from the vertex to the midpoint

𝟐 𝟑 QC = QZ QC = 2CZ

PY = 21 PC = CY = 8 12 4 8 8 6

Page 165 16 4

MID-SEGMENTS OF A TRIANGLE A mid-segment of a triangle connects the midpoints of two sides of the triangle.

Mid-segment Theorem The midsegment of a triangle is parallel to the third side and is half as long as that side. D B C E A

Identify the 3 pairs of parallel lines shown above

The mid-segment of a triangle is parallel to the third side and is half as long as that side. y y 2x x z z

2b. 2a.

Example 1 In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU. 5 ft 16 ft TU = ________ PR = ________

Example 3 In the diagram, ED and DF are midsegments of triangle ABC. Find DF and AB. 3X – 4 5X+2 2 (DF ) = AB 2 (3x – 4 ) = 5x + 2 6x – 8 = 5x + 2 x – 8 = 2 x = 10 x = ________ 10 DF = ________ 26 AB = ________ 52

ED is a mid-segment of ABC

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