5-1: Special Segments in Triangles

5-1: Special Segments in Triangles • Expectation: • G1.2.5: Solve multi-step problems and construct proofs about the properties of medians, altitudes and perpendicular bisectors to the sides of a triangle and the angle bisectors of a triangle. 5-1: Special Segments in Triangles

You have a piece of string 120 cm long. What is the area of the largest square you can enclose? 5-1: Special Segments in Triangles

What is the length of the hypotenuse of the isosceles triangle below? • 20 • 40 • 800 • 20√2 • 40√2 20 5-1: Special Segments in Triangles

Median of a Triangle Defn: Median of a Triangle: A segment is a median of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is the midpoint of the side opposite that vertex. 5-1: Special Segments in Triangles

Medians of a Triangle Every triangle has 3 medians. 5-1: Special Segments in Triangles

Centroids • The medians of a triangle will always intersect at the same point - the centroid. The centroid of a triangle is located 2/3 of the distance from the vertex to the midpoint of the opposite side. 5-1: Special Segments in Triangles

Centroid centroid 5-1: Special Segments in Triangles

Centroid 5-1: Special Segments in Triangles

Points U, V, and W are the midpoints of YZ, XZ and XY respectively. Find a,b, and c. 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle Defn: Perpendicular Bisector of a Triangle: A segment is a perpendicular bisector of a triangle iff it is the perpendicular bisector of a side of the triangle. 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle Every triangle has 3 perpendicular bisectors. 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle 5-1: Special Segments in Triangles

The 3 perpendicular bisectors of any triangle will intersect at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter and is the center of a circle that contains all 3 vertices of the triangle. 5-1: Special Segments in Triangles

Perpendicular Bisector Theorem A point lies on the perpendicular bisector of a segment iff it is equidistant from the endpoints of the segment. 5-1: Special Segments in Triangles

C is on the perpendicular bisector of AB. Perpendicular Bisector Theorem If AC = BC, then C A B 5-1: Special Segments in Triangles

If l is the perpendicular bisector of AB, C A B D l Perpendicular Bisector Theorem then AC = BC and AD = BD. 5-1: Special Segments in Triangles

Lines s, t,and u are perpendicular bisectors of ∆FGH and meet at J. If JG = 4x + 3, JH = 2y - 3, JF = 7 and HI = 3z - 4, find x, y,and z. t H s u G I J 1 1 F 5-1: Special Segments in Triangles

Altitudes of Triangles Defn: Altitude of a Triangle: A segment is an altitude of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is on the line containing the opposite side such that the segment is perpendicular to line. 5-1: Special Segments in Triangles

Altitudes of Triangles Every triangle has 3 altitudes that will always intersect in the same point. 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is acute, then the altitudes are all in the interior of the triangle. 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle. 5-1: Special Segments in Triangles

Altitudes of Triangles If the triangle is an obtuse triangle, then one altitude is in the interior and the other 2 altitudes are in the exterior of the triangle. 5-1: Special Segments in Triangles

Altitudes of Triangles 5-1: Special Segments in Triangles

ZC is an altitude, m∠CYW = 9x + 38 and m∠WZC = 17x. Find m∠WZC. Z A X W C Y 5-1: Special Segments in Triangles

Angle Bisectors of Triangles Defn: Angle Bisector of a Triangle: A segment is an angle bisector of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is any other point on the triangle such that the segment bisects an angle of the triangle. 5-1: Special Segments in Triangles

Angle Bisectors of Triangles Every triangle has 3 angle bisectors which will always intersect in the same point - the incenter. The incenter is the same distance from all 3 sides of the triangle. The incenter of a triangle is also the center of a circle that will intersect each side of the triangle in exactly one point. 5-1: Special Segments in Triangles

Angle Bisectors of Triangles 5-1: Special Segments in Triangles

RU is an angle bisector, m∠RTU = 13x – 24, m∠TRS = 12x – 34 and m∠RUS = 92. Determine m∠RSU. Is RU ⊥ TS? R S T U 5-1: Special Segments in Triangles

Angle Bisector Theorem If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. 5-1: Special Segments in Triangles

Angle Bisector Theorem A If D is on the bisector of ∠ABC, then X D B Y C DX = DY. 5-1: Special Segments in Triangles

Angle Bisector Converse Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle. 5-1: Special Segments in Triangles

Angle Bisector Converse Theorem X If WX = WY, then W is on the bisector of ∠XYZ. W Y Z 5-1: Special Segments in Triangles

In ΔABC below, AB ≅ BC and AD bisects ∠BAC. If the length of BD is 3(x + 2) units and BC = 42 units, what is the value of x? • 5 • 6 • 12 • 13 A B C D 5-1: Special Segments in Triangles

Assignment • Worksheet 5-1 5-1: Special Segments in Triangles

5-1: Special Segments in Triangles