Unit 5 PPT

UNIT 5Relationships Within Triangles

Do Now: Throwback. Find the measurement of each unknown angle.

NAME: DATE: JANUARY 7, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: MIDSEGMENT

Aim: How can we use properties of midsegments to solve problems?

Homework: Worksheet Due Monday 1/11/16

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TRIANGLE MIDSEGMENT THEOREM Midsegment- a segment connecting the

midpoints of two sides of a triangle.

EXAMPLE 1 What are three pairs of parallel

segments in ∆DEF?

EXAMPLE 2 In ∆QRS, T, U and B are midpoints. What

are the lengths of lines TU, UB and QR?

EXAMPLE 3 CD is a bridge being built over a lake, as

shown in the figure below. What is the length of the bridge?

INDEPENDENT PRACTICE

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Do Now: Find distance across each lake.

NAME: DATE: JANUARY 8, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: PERPENDICULAR BISECTOR THEOREM

Aim: How do we use the properties of perpendicular bisectors?

Homework: Worksheet Due Monday 1/11/16

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PERPENDICULAR BISECTOR THEOREM Equidistant – when one or more objects

are the same distance away from another object.

If a point is on the perpendicular bisector of a segment, then it equidistant from the endpoints of the segment.

CONVERSE OF THE PERPENDICULAR BISECTOR THEOREM

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

EXAMPLE 1 What is the length of line AB?

EXAMPLE 2 A park director wants to build a T-shirt

stand equidistant from the Rollin’ Coaster and the Spaceship Shoot. What are the possible locations of the stand?

INDEPENDENT PRACTICE

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Do Now: 1) Hand in your homework2)

NAME: DATE: JANUARY 11, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: ANGLE BISECTOR THEOREM

Aim: How do we use the properties of angle bisectors?

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Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/19/16

ANGLE BISECTOR THEOREM If a point is on the bisector of an angle,

then the point is equidistant from the sides of the angle.

CONVERSE OF THE ANGLE BISECTOR THEOREM If a point in the interior of an angle is

equidistant from the sides of the angle, then the point is on the angle bisector.

EXAMPLE 1 What is the length of line RM?

EXAMPLE 2 What is the length of FB?

INDEPENDENT PRACTICE1.

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Do Now: Throwback! Using the compass and straightedge, construct a perpendicular bisector.

NAME: DATE: JANUARY 12, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: PERPENDICULAR BISECTORS

Aim: How can we identify properties of perpendicular bisectors?

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Homework: Worksheet Due Monday 1/19/16. Benchmark 1/19/16

VOCABULARY Concurrent – three or more lines

intersect at one point Point of concurrency – the point at which

concurrent lines intersect Circumcenter – point of concurrency of

perpendicular bisectors Circumscribed – when a circle surrounds

another shape by touching all the vertices of the shape.

CONCURRENCY OF PERPENDICULAR BISECTORS THEOREM

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices

HOW TO CONSTRUCT PERPENDICULAR BISECTOR POINT OF CONCURRENCY Draw ∆ABC Construct a perpendicular bisector of

line AB Construct a perpendicular bisector of

line BC Construct a perpendicular bisector of

line AC Label the point of intersection as P.

CONCURRENCY OF PERPENDICULAR BISECTORS THEOREM

The circumcenter of a triangle can be inside, on or outside a triangle.

EXAMPLE 1 What are the coordinates of the

circumcenter of the triangle with vertices P(0,6), O(0,0), and S(4,0)?

EXAMPLE 2 A town planner wants to locate a new

fire station equidistant from the elementary, middle and high schools. Where should she locate the station?

INDEPENDENT PRACTICE Construct perpendicular bisector

concurrencies of a:Acute triangleRight triangleObtuse triangle

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Do Now: Throwback: Bisect an acute angle and an obtuse angle.

NAME: DATE: JANUARY 13, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: ANGLE BISECTORS

Aim: How can we identify properties of angle bisectors?

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Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16

VOCABULARY Incenter – point of concurrency of angle

bisectors of a triangle Inscribed – when the largest possible

circle is inside a shape.

CONCURRENCY OF ANGLE BISECTORS THEOREM The bisectors of the angles of a triangle

are concurrent at a point equidistant from the sides of the triangle.

EXAMPLE 1 GE = 2x – 7 and GF = x + 4. What is

GD?

EXAMPLE 2 Name the point of concurrency of the

angle bisectors

INDEPENDENT PRACTICE1. Construct the incenter of:

An acute triangleA right triangle An obtuse triangle

2. Find the value of x

3. Bonus: Find the circumcenter of ∆ABC:1. A(5,2), B(-1,2), C(-1,-3)2. A(2,-2), B(-4,-2), C(-4, -7)

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Do Now: Town officials want to place a recycling bin so that it is equidistant from the lifeguard chair, the snack bar and the volleyball court. Where should they place it?

NAME: DATE: JANUARY 14, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: MEDIANS AND ALTITUDES

Aim: How can we identify properties of medians and altitudes of a triangle?

Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16

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VOCABULARY Median of a triangle – a

segment whose endpoints are a vertex and a midpoint of the opposite side

Altitude of a triangle – the perpendicular segment from a vertex of a triangle to the line containing the opposite side.

CONCURRENCY OF MEDIANS THEOREM The medians of a triangle are concurrent

at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. The point where the lines meet is called the centroid of the triangle.

CONCURRENCY OF ALTITUDES THEOREM The lines that contain the altitudes of a

triangle are concurrent. The point where the three altitudes meet is called the orthocenter of the triangle. The orthocenter could be inside, on or outside the triangle.

EXAMPLE 1 In the diagram below, XA = 8. What is

the length of XB?

EXAMPLE 2 For ∆PQS, is PR a median, altitude or

neither? Explain Is QT a median, altitude or neither?

Explain

INDEPENDENT PRACTICETim

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Do Now: Algebra Throwback!Solve the following inequalities:

NAME: DATE: JANUARY 15, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: INEQUALITIES IN ONE TRIANGLE

Aim: How can we use inequalities involving angles and sides of triangles?

Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16

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TRIANGLE INEQUALITIES

EXAMPLE 1 A town park is triangular. A landscape

architect wants to place a bench at the corner with the largest angle. Which two streets form the corner with the largest angle?

EXAMPLE 2 List the sides of ∆TUV in order from

shortest to longest.

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:

EXAMPLE 3 Can a triangle have sides with the given

lengths? Explain.1. 3 ft, 7 ft, 8 ft

2. 5 ft, 10 ft, 15 ft

EXAMPLE 4 Two sides of a triangle are 5 ft and 8 ft

long. What is the range of possible lengths for the third side?

INDEPENDENT PRACTICETim

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Do Now: Algebra Throwback!Solve the following equation using PEMDAS

-3 * ( 5x + 8 ) - 22 / 4 + 3x

NAME: DATE: JANUARY 22, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: POINTS OF CONCURRENCY

Aim: How can we review points of concurrency?

Homework: Come up with your own way to remember points of concurrency

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HOW TO REMEMBER POINTS OF CONCURRENCY All Of : Altitudes - Orthocenter My Children: Medians - Centroid Are Bringing In: Angle Bisectors - Incenter Peanut Butter Cookies: Perpendicular Bise

ctors - Circumcenter

Do Now: In the diagram, the perpendicular bisectors (shown with dashed segments) of MNP meet at point O—the circumcenter. Find the indicated measure.

NAME: DATE: JANUARY 25, 2016 UNIT: RELATIONSHIPS WITHIN TRIANGLESTOPIC: POINTS OF CONCURRENCY

Aim: How can we review points of concurrency?

Homework: Pass your regents

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1. MO = ___________ 2. PR = __________ 3. MN = __________ 4. SP = __________ 5. mMQO = __________ 6. If OP = 2x, find x.

EXAMPLEPoint S is the centroid of RTW, RS = 4, VW = 6,

and TV= 9. Find the length of each segment.

RV = __________ SU = __________ RU = __________ RW = __________ TS = __________ SV = __________