Relationships Within Triangles Chapter 5

5.1 – Midsegment Theorem Midsegment of a triangle Segment that connects midpoints of two sides of a triangle Every triangle has three midsegments

Midsegment Theorem Theorem 5.1 – 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 side 𝐷𝐸 || 𝐴𝐶 𝑎𝑛𝑑 𝐷𝐸= 1 2 𝐴𝐶 d E c B A

5.2 – Use Perpendicular Bisectors Segment, ray, line, or plane that is perpendicular to a segment at its midpoint Equidistant Points on the perpendicular bisector of a segment are the same distance from the segment’s endpoints

Theorems Theorem 5.2 – 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 Theorem 5.3 – Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment

Example 1 Line BD is the perpendicular bisector of segment AC. Find the length of AC. C B D A 3x + 14 5x

Concurrency of Lines, Rays & Segments Concurrent When three or more lines, rays, or segments intersect in the same point Point of concurrency Point of intersection of three or more lines, rays, or segments

Theorem 5.4 Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle (If EG, EF, and ED are perp. Bisectors, then AG=BG=CG)

Example 3 Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor (D) Find a location for the distributor that is equidistant from the three carts (you will draw on the next slide)

Circumcenter Circumcenter Activity: Point of concurrency of the three perpendicular bisectors of a triangle Activity: Construct the perpendicular bisectors of a triangle, and check to see if a circle can be constructed so that all vertices of the triangle lie on the circle

5.3 – Use Angle Bisectors of a Triangle Remember: An angle bisector is a ray that divides an angle into two congruent adjacent angles The distance from a point to a line is the length of the perpendicular segment from the point to the line

Theorems Theorem 5.5 – Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle Theorem 5.6 – Converse of Angle Bisector Thrm. 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

Examples Example 1 P. 312 Example 2 P. 313

Example 3 For what value of x does P lie on the bisector of <A? GP #1-4 p. 313 B x + 3 P A 2x – 1 C

Incenter of a Triangle Theorem 5.7 – 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 Point of concurrency of the three angle bisectors of a triangle Always lies inside the triangle Example 4, p. 314

5.4 – Use Medians & Altitudes Median of a triangle Segment from a vertex to the midpoint of the opposite side Centroid Point of concurrency of the three medians of a triangle

Theorem Theorem 5.8 – Concurrency of Medians of Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The medians of PQR meet at point V 𝑃𝑉= 2 3 𝑃𝑇, 𝑄𝑉= 2 3 𝑄𝑈 𝑅𝑉= 2 3 𝑅𝑆

Altitudes of a Triangle Altitude of a triangle Perpendicular segment from a vertex to the opposite side of to the line that contains the opposite side Theorem 5.9 – Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent Orthocenter Point at which the lines containing the three altitudes of a triangle intersect

5.5 – 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 Theorem 5.11 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

Triangle Inequality Theorem Theorem 5.12 – 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 3 A triangle has one side of length 12 and another length 8. Describe the possible lengths of the third side. GP #3 p.332