04/21/23 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.

You have a piece of string 120 cm long. What is the area of the largest square you can enclose?

04/21/23 5-1: Special Segments in Triangles

What is the length of the hypotenuse of the isosceles triangle below?

a. 20b. 40c. 800d. 20√2e. 40√2

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20

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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.

04/21/23 5-1: Special Segments in Triangles

Medians of a TriangleEvery triangle has 3 medians.

04/21/23 5-1: Special Segments in Triangles

Medians of a Triangle

Every triangle has 3 medians.

04/21/23 5-1: Special Segments in Triangles

Medians of a Triangle

Every triangle has 3 medians.

04/21/23 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.

04/21/23 5-1: Special Segments in Triangles

Centroid

centroid

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Centroid

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Points U, V, and W are the midpoints of YZ, XZ and XY respectively. Find a, b, and c.

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Perpendicular Bisectors of a TriangleDefn: 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.

04/21/23 5-1: Special Segments in Triangles

Perpendicular Bisectors of a TriangleEvery triangle has 3 perpendicular bisectors.

04/21/23 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle

04/21/23 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle

04/21/23 5-1: Special Segments in Triangles

Perpendicular Bisectors of a Triangle

04/21/23 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.

04/21/23 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.

04/21/23 5-1: Special Segments in Triangles

Perpendicular Bisector Theorem

A

C

B

If AC = BC, then

C is on the perpendicular bisector of AB.

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Perpendicular Bisector Theorem

A B

l

C

D

If l is the perpendicular bisector of AB,

then AC = BC and AD = BD.

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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.

H

G

F

s

t

u

I

11J

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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.

04/21/23 5-1: Special Segments in Triangles

Altitudes of Triangles

Every triangle has 3 altitudes that will always intersect in the same point.

04/21/23 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.

04/21/23 5-1: Special Segments in Triangles

Altitudes of TrianglesIf the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle.

04/21/23 5-1: Special Segments in Triangles

Altitudes of TrianglesIf 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.

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Altitudes of Triangles

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ZC is an altitude, m CYW = 9x + 38 and ∠m WZC = 17x. Find m WZC.∠ ∠

Y

X

Z

W

A

C

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Angle Bisectors of TrianglesDefn: 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.

04/21/23 5-1: Special Segments in Triangles

Angle Bisectors of TrianglesEvery 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.

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Angle Bisectors of Triangles

04/21/23 5-1: Special Segments in Triangles

Angle Bisectors of Triangles

04/21/23 5-1: Special Segments in Triangles

Angle Bisectors of Triangles

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RU is an angle bisector, m RTU = 13x – 24, ∠m TRS = 12x – 34 and m RUS = 92. Determine m RSU. ∠ ∠ ∠Is RU TS? ⊥

R

T

S

U

04/21/23 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.

04/21/23 5-1: Special Segments in Triangles

Angle Bisector Theorem

If D is on the bisector of ∠ABC, then

A

Y

X

D

C

B

DX = DY.

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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.

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Angle Bisector Converse Theorem

X

Z

YW

If WX = WY, then W is on the bisector of ∠XYZ.

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?

A. 5B. 6C. 12D. 13

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A

CDB

04/21/23 5-1: Special Segments in Triangles

Assignment

Worksheet 5-1