Median ~ Hinge Theorem

_____(0-10 pts.) Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a trian-gle theorem. Give at least 3 examples of each.

Describe what a median is.

•Median: Is a line drawn from one vertex of a triangle to the midpoint of the opposite side.

•Every triangle has three medians.•Medians are concurrent

Examples:

Explain the concurrency of medians of a triangle theorem.•The point of concurrency of the medians is

called the centroid. (Where all median meet.)

•Centroid Theorem: on every median of any triangle, the distance from the vertex to the centroid is

double the distance from the centroid to the midpoint of the opposite side.

Examples:

_____(0-10 pts.) Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples.

Describe what an altitude of a trian-gle is.•Altitude: A line from the vertex perpendic-

ular to the opposite side.•An altitude can be inside, outside, or on

the triangle.

Examples:

Explain what an orthocenter is.

•Orthocenter: The point of congruency for the 3 altitudes. (where all altitudes meet)

•The orthocenter of an acute triangle is in-side the triangle. The orthocenter of a right triangle is on the vertex of the 90 degrees angle. The orthocenter of an ob-tuse triangle is on the outside of the trian-gle.

Examples:

Explain the concurrency of altitudes of a triangle theorem.•That the three altitudes of any triangle are

concurrent. (this point of concurrency is called orthocenter)

_____(0-10 pts.) Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples.

Describe what a midsegment is.•A segment that joins any two midpoints of

a triangle. The midsegment is parallel to the opposite side, and it is half as long as the opposite side.

Examples:

Explain the midsegment theo-rem.•A midsegment of a triangle is parallel to a

side of the triangle.• Its length is the half the length of that

side.

Examples:

_____(0-10 pts.) Describe the relationship between the longer and shorter sides of a triangle and their opposite an-gles. Give at least 3 examples.

Angle-Side Relationships in Tri-angles•The side that is opposite the largest angle

will always be the longest side.•The side that is opposite the smallest an-

gle will be always the shortest side.

Examples:

_____(0-10 pts.) Describe the exterior angle inequality. Give at least 3 examples.

Exterior angle inequality.

•An exterior angle of a triangle that is greater than either of the non-adjacent in-terior angles.

Examples:

_____(0-10 pts.) Describe the triangle inequality. Give at least 3 examples.

Triangle inequality

• In every triangle the sum of the lengths of the two shorter sides must always be longer than the 3rd side.

Examples:

_____(0-10 pts.) Describe how to write an indirect proof. Give at least 3 examples.

Indirect Proof

•You begin by assuming that the conclusion is false.

•Then show that this assumption leads to a contradiction.

•Also called proof by contradiction.

Examples:

_____(0-10 pts.) Describe the hinge theorem and its con-verse. Give at least 3 exam-ples.

Hinge theorem and its con-verse.• If the two sides of two triangles are con-

gruent but the third side is not congruent, then the triangle with the longer side, will have a larger included angle.

•Converse: If two sides of one triangle are congruent to two sides of another triangle and the tird sides are not congruent, then the larger included angle is across from the longer third side.

Examples: