CHAPTER 5 Relationships within Triangles By Zachary Chin and Hyunsoo Kim

THIS is a triangle. For those of you who are unaware…

How to draw a Triangle Step 1: Draw a line segment. Step 2: Draw another line segment with a common endpoint. Step 3: Connect with another line segment.

NO DUH!

5.1: Midsegments and Coordinate Proofs

Midsegments A midsegment is a line in a triangle that connects the midpoints of two sides of a triangle. Every triangle has three midsegments.

Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that third side. > > A B D C E DE AC and DE=½AC

Example #1 Find the length of the midsegment (DE is a midsegment) A B C D E 20 DE = 10 DE = ½AC, so…

Coordinate Proofs A coordinate proof involves placing geometric figures in a coordinate plane. When you use variables to represent the coordinates of a figure in a coordinate proof, the results are true for all figures of that type.

Coordinate Proofs Triangle: (a, b) (0, 0) (c, 0) y x

Example #2 Place a regular octagon in the coordinate plane. (0, a) (b, 0)(c, 0) (d, a) (0, e) (d, e) (c, f)(b, f) y x

5.2: Use Perpendicular Bisectors

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. A point is equidistant from two figures if the point is the same distance from each figure.

Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A B C P If CP is the perpendicular bisector of AB, then CA = CB.

Converse of the Perpendicular Bisector Theorem A B C P D In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB.

A B C P CP is the perpendicular bisector of AB. Find CB. Example #3 10x 2 +15x-6 -6x+4 CB = 1.6 units Use CA = CB, so… 10x+15x-6 = -6x+4 Solve for x, plug it into -6x+4…

When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments. The point of intersection of the lines, rays, or segments is called the point of concurrency.

Concurrency of Perpendicular Bisectors Theorem The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle. circumcenter circumscribed circle

5.3: Use Angle Bisectors of Triangles

Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Converse of the Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. incenter inscribed circle

Example #4 3x+2 6x-4 Find the value of x. x = 2 BD = CD, so… 6x-4 = 3x+2

5.4: Use Medians and Altitudes

A median of a triangle is a segment from a vertex to the midpoint of the opposite vertex. The point of concurrency, called the centroid, is inside the triangle.

Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side. For example, AG = 2 / 3 AM A centroid

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

Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent. orthocenter

Example #5 If AG = 2x+3, and AM A = 6x+6, find the value of x. AG = 2 / 3 AM A 2x+3 = 2 / 3 (6x+6) x = -½

5.5 Use Inequalities in a Triangle

Theorem 5.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side This angle is larger than… This one

Theorem 5.11 A B C If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. AC < BC because m ∠ A is greater than m ∠ B.

Example #6 Given that m ∠ A is greater than m ∠ C, which side is longer, AB or BC? The answer is BC.

The Triangle Inequality Not all triangles can be made with any lengths. If you have a “triangle” that has sides of 2, 2, and 4, this triangle is not possible to draw. 4=4, as 2+2=4, so that means that this triangle will become a straight line. This “triangle inequality” leads to…

Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example #7 Is it possible to make a triangle with sides of 43.6, 57.2, and 101.4? No, because ( ) is less than

5.6: Inequalities in Two Triangles

Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle. Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Example #8 5 3 Given BC and EF, which is larger, ∠ A or ∠ D? ∠A∠A Because 5 > 3.