Warm-Up. Centroid and Orthocenter. Lesson Question. Lesson Goals. Distinguish between the centroid and the. Prove the. Use the

Warm-Up

Lesson Goals

Centroid and Orthocenter

Lesson Question

?

Analyze intersections of in triangles

and intersections of in triangles.

Distinguish between the centroid and the

.

Use the

ratio theorem to solve problems.

Prove the

of a triangle meet at a point.

Warm-Up

W

K Words to Know

Fill in this table as you work through the lesson. You may also use the glossary to help you.

altitude

in a triangle, a segment from a vertex to the opposite side or to the line containing the

side

centroid

the point of of the three of a triangle

median

a segment from one of a triangle to the midpoint of the side

orthocenter the point of concurrency of the three

containing the of a triangle

point of concurrency

the at which or more lines, rays, or line segments

Centroid and Orthocenter

• A of a triangle is a segment from one vertex of a triangle to the

of the opposite side.

Warm-Up

Altitudes and Medians

Centroid and Orthocenter

• An of a triangle is a segment from a vertex to the opposite

side that is to the side, or the height of a triangle.

C

A

B

C

A

B

The centroid is where the of a triangle intersect.

A point of concurrency is a point where three or more lines .

2

Slide

Points of Concurrency

The orthocenter is where the of the triangle intersect.

Orthocenters and Centroids

Instruction Centroid and Orthocenter

Point of

C

A

B P

D

E

F

The centroid is the center of gravity for any triangle. All centroids balance the triangle at a single point. This means all centroids are located of a triangle.

Z U

W

V

R

T S

BG =

4

Slide

The location of the depends on the type of triangle.

Locations of Orthocenters

The Centroid Ratio Theorem

Instruction Centroid and Orthocenter

Inside the triangle for an triangle

the triangle for an obtuse triangle

On the triangle for a triangle O

C A

B

B O

A

C

O B A

C

Centroid ratio theorem: Along each median

in a triangle, the distance between the and the centroid is the distance

between the centroid and the side the vertex.

G

D F E

A

B C

= 2GD

= 2FG

4

Slide

Example: Q is the centroid of triangle JKL. What is the length of segment QM?

Using the Centroid Ratio Theorem

Instruction Centroid and Orthocenter

3𝑥 − 4 4𝑥 Q

P M

N J

K KQ = 2QM L

4𝑥 = 2(2𝑥 − 4) 4𝑥 = 6𝑥 − 8

8 =

= 𝑥 QM

3(4) − 4 = 12 − 4

= QM = KQ =

Plan for proving that all the medians of a triangle meet at a point:

• Segments BD and CE are

of triangle ABC.

• Write linear equations for BD and CE.

• Solve for the intersection point, G.

• Write the of AG.

• Write an expression for the midpoint of BC, F.

• Show that AF is the of BC.

Proving the Medians of a Triangle Meet at a Point: The Plan

𝑦

𝑥 G

D(𝑐, 0) F E(𝑎, 𝑏)

B(2𝑎, 2𝑏)

A(0, 0) C(2𝑐, 0)

7

7

Slide

Proving the Medians of a Triangle Meet at a Point: Finding Slopes

If segments BD and CE are medians of triangle ABC, show that all the medians of a triangle meet at a point.

• Find the slopes of BD and CE.

Slope of BD:

Proving the Medians of a Triangle Meet at a Point: Writing Equations

Instruction Centroid and Orthocenter

𝑚 =2𝑏 − 0 2𝑎 − 𝑐 =

Slope of CE:

𝑚 = 𝑏 − 0 𝑎 − 2𝑐 =

𝑦

𝑥 G

D(c, 0) E(a, b) F

B(2a, 2b)

A(0, 0) C(2c, 0)

• Use the to write equations in point-slope form, and then solve for y.

• Use the point (2c, 0) as (𝑥₁, 𝑦₁).

BD: m = 2b

2a⁻c CE: m = b a⁻2c

CE: y − 0 = ba−2c (𝑥 − 2c)

y =

𝑏

𝑎 − 2𝑐 𝑥 −

𝑦

𝑥 G

D(c, 0) E(a, b) F

B(2a, 2b)

A(0, 0) C(2c, 0)

9

Slide

• Find the intersection point of the lines.

Proving the Medians of a Triangle Meet at a Point: Finding a Point of Intersection

Instruction Centroid and Orthocenter

BD: y = 2b

2a⁻c 𝑥 − 2bc 2a⁻c CE: y = b

a−2c 𝑥 − 2bc

a−2c G

D(c, 0) E(a,b) F

B(2a, 2b)

A(0, 0) C(2c, 0)

2b

2a − c 𝑥 − 2bc

2a − c = b

a − 2c 𝑥 − 2bc a − 2c 2b𝑥 −2bc

2a⁻c =b𝑥 −2bc a⁻2c

(a − 2c)(2b𝑥 − 2bc) = (2a − c)(b𝑥 − 2bc)

2ab𝑥 − 2abc − 4bc𝑥 + 4bc ²= 2ab𝑥 − 4abc − bc𝑥 + 2bc ²

−2abc − 4bc𝑥 + 4bc ²= − 4abc − bc𝑥 + 2bc ²

𝑥 = 2abc + 2bc ²= 2a + 2c =

( , )

2a + 2c

2 , 2b + 0

2 =

11

Slide

G 2(a + c) 3 ,2b

3 is the intersection point.

• Write the equation of AG.

Proving the Medians of a Triangle Meet at a Point: Writing the Equation of the Potential Median

Instruction Centroid and Orthocenter

𝑦

𝑥 G

D(c, 0) E(a, b) F

B(2a, 2b)

A(0, 0) C(2c, 0)

Equation of AG: = ba+c 𝑥 Slope of AG: m =

2b3 2(a + c)

3

=

• Find the midpoint of BC

Proving the Medians of a Triangle Meet at a Point: Finding a Midpoint

𝑦

𝑥 G

D(c, 0) E(a, b) F

B(2a, 2b)

A(0, 0) C(2c, 0)

13

Slide

Proving the Medians of a Triangle Meet at a Point

Instruction Centroid and Orthocenter

𝑦

𝑥 G

D(c, 0)

F(a + c, b) E(a, b)

B(2a, 2b)

A(0, 0) C(2c, 0)

G 2(a + c) 3 ,2b Plan for proving that all the medians of a 3

triangle meet at a point:

• Segments BD and CE are medians of triangle ABC.

• Write linear equations forBD and CE.

• Solve for the intersection point, .

• Write the equation of AG.

• Write an expression for the midpoint of BC, F.

• Show that is the median of

.

Summary

Answer

?

Slide

2

The of a triangle is where the altitudes meet.

• The orthocenter is inside the triangle for an , outside for an obtuse triangle, and on for a right triangle.

The of a triangle is where the medians meet.

• The distance between the and the centroid is twice the distance between the centroid and the

Review: Key Concepts Lesson

Question

Centroid and Orthocenter

G E

D F

A

B C

C A

B D O

E

F orthocenter

centroid

Summary

Use this space to write any questions or thoughts about this lesson.

Centroid and Orthocenter