Ch. 11 - Similarity Class Notes. What Is Similarity?

transcript

Ch. 11 - Similarity

Class Notes

What Is Similarity?

Warm-Up

• Each pair, open textbook to p. 564• Each person use patty paper,

protractor and ruler to complete steps 1 & 2. Mark corners, A, B etc.• Mark all measures of sides and angles

on patty paper.• Glue paper into your notebook. In ten

minutes, you will be quizzed on this.

Quiz Question 1• Whiteboard, marker & rag in table.• Hold up answer, facing forward, 1 min.Q: Are these shapes congruent?

A. Yes, they have the same angles.B. Yes, they can be superimposed.C. No, they have the same angles but

not the same sides.D. Cannot be determined. (CBD)

NotesPolygons are similar if,• Their corresponding angles are

congruent (same) and,• Their corresponding sides are

proportional. That is, you can multiply each side by the same scale factor to get the corresponding side length.

Warm-Up, Part II (5 minutes)• Calculate the ratio of each pair of

corresponding sides in these shapes. Record on patty paper.• Example: RS/CD = 4.1cm/2.3cm

ratio RS/CD = 1.8• In five minutes, you will be quizzed.

Quiz Question 2• Whiteboard, marker & rag in table.• Hold up answer, facing forward, 1 min.Q: Are these shapes similar?

A. Yes, they can be superimposed.B. No, side lengths not all the same.C. Yes, they have the same angles and all

pairs of corresponding sides are proportional.

D. Cannot be determined. (CBD)

Quiz Question 3• Whiteboard, marker & rag in table.• Hold up answer, facing forward, 5 min.–Draw three similar, but not congruent

triangles.–Mark all side lengths.–Mark which angles & sides are

congruent.–Draw one non-similar triangle with side

lengths.

Notes• Dilation – Process of growing or

shrinking. Example: Eye pupils dilate in dark.

• Scale factor – How much the pre-image has grown/shrunk to create the image. Example: 3:1 = scale factor 3• Calculate scale factor by dividing

length of corresponding sides:image/pre-image.

• Calculate scale factor by dividing length of corresponding sides image/pre-image.• Pre-image 3 cm -> Image 9 cm

= 9/3 = scale factor 3• Pre-image 8 cm -> Image 2 cm

= 2/8 = scale factor ¼ = 0.25• Scale factor > 1 is growing ; < 1 is

shrinking

Polygons are similar if,• Their corresponding angles are

congruent (same) and,• Their corresponding sides are

proportional, have the same scale factor one shape to the next.• The ratio of sides within each shape

is the same: small-to-medium-to-big.

Quiz Question 1 (1 minute)

Q: Are these shapes,A. CongruentB. SimilarC. Congruent & SimilarD. Neither

• Are corresponding angles congruent? YES. (90o)

• Are corresponding sides proportional? NO.

• Conclusion – These polygons are not similar or congruent.

Quiz Question 2

• Answer # 1, p. 568 on whiteboard.

Quiz Question 3

• Answer # 8, p. 568 on whiteboard.

Quiz Question 4

• Answer # 5, p. 568 on whiteboard.• Also write the scale factor.

Cleanup Grade

• Start at 100% each quarter.• Keep credit if:– Push in chair to touch table.– Pick up trash near desk, even if not yours.– Remove trash inside desk.– Erase whiteboard.– Return textbook, whiteboard, marker, rag into

desk.• Also start with 100% for following rules…

• HW p. 568 # 1-10

• # 1 & 2 Find two shapes that have same angles and corresponding parts, but have different sizes.• # 3 – 5 Draw original shape on graph

paper. Then repeat but make all sides bigger by same amount. (2x?)

Drawing Dilations

Warm-Up1. Place a point on the left of a new page in your

notebook.2. Draw three rays ‘radiating’ out from this

point, to the upper, middle & lower right.3. Randomly place an additional point on each

ray. Connect them to form a triangle.4. Place a second point on each ray, at twice the

distance as the first. Connect to make a second triangle.

5. What do we notice about the two triangles?

Q: What is the relationship between these triangles?

A. CongruentB. SimilarC. Congruent & SimilarD. Neither

Q: What is the scale factor?A. 0.5B. 1.0C. 2.0D. None of the above

Q: Corresponding sides in these two triangles are:

A. congruentB. parallelC. perpendicularD. None of the above

Dilations Centered on Same Point

Dilations Centered on Same PointDilations work for all polygons…

Center of dilation can be inside shapes.

Scale factor can be negative…

What is the scale factor in this dilation?

Similar Shapes Centered on Same Point of Dilation...

• Corresponding angles are congruent.• Corresponding sides are parallel .• Every pair of corresponding sides has the same

ratio to each other.• Ratios of center-to-vertex distances are the

same for corresponding vertices in each shape.

• Ratios between sides within the same shape, are identical for pre-image and image.

How to Know ifShapes are Similar

& Prove It?

• Have HW open in first minute of class to earn credit.• Open textbook to p. 568.

• You will be quizzed on the following graph in 10 minutes.• On your personal whiteboard graph:

(3 ,2) (4, -2) (-2, -3)• Dilate triangle with scale factor of 2,

Draw rays centered on origin.• Write a coordinate rule for this.

Warm-Up

Similar Shapes• Corresponding angles are congruent.• Corresponding sides are parallel (if share the same center of dilation.)• Pairs of corresponding sides have the

same ratio to each other, old & new… AB to A’B’; BC to B’C’…

Scale Factor = new length/old lengths

Transformation rules for dilationsonly when centered on the origin (0,0)• 2x dilation: (x, y) → (2x, 2y)• 1/3x dilation:(x, y) → (1/3x, 1/3y)• Stretch - shapes not similar!

(x, y) → (2x, y) Growing wider only.• Another stretch:

(x, y) → (2x, 3y) Growing differentlyin x & y directions.

Similar Shapes

• Ratios between sides within the same shape, small:medium:large, are identical for pre-image and image.• Ratios of center-to-vertex distances

are the same for corresponding vertices in each shape.

• Are corresponding angles congruent? YES. (90o)• Are corresponding sides proportional? NO.• Conclusion – These polygons are not similar or

congruent.

If all corresponding angles remain congruent, must sides be proportional?

If all sides are proportional, must corresponding angles remain congruent?

• Scale factor consistently 18/12=1.5• But…are corresponding angles

congruent? NO!

Are these Shapes Similar?

• Corresponding angles congruent?• Corresponding sides proportional?• Conclusion?

Are these Shapes Similar?

• Corresponding angles congruent? YES.• Corresponding sides proportional? YES.• Conclusion – These polygons are

similar. CORN ~ PEAS

For Triangles only…Angle-Angle Triangle Similarity Shortcut

• If any two angles in two triangles are congruent…then both triangles must be similar! (AA Shortcut)

• Do not need to measure third pair of angles (Triangle Sum Conjecture says they must be the same.)

• Do not need to measure sides or calculate scale factor… sides must all grow/shrink by same ratio to keep angles congruent.

• p. 574 # 1-10

How to Indirectly MeasureHeights of Tall Objectswith Similar Triangles

• You will measure the height of the…1. Gold ball on flagpole outside2. Ceiling by vending machines3. Bottom edge of U.S. flag in

commons4. Water tower (tricky; can’t measure

directly to base…use similar triangles, twice?)

Indirectly Measure Heights of Tall Objects with Similar Triangles

Method #1The Mirror Method

• String, knots 1 meter apart• 30 cm ruler• Mirror with taped edges• When done, leave all supplies neatly

on desk• You assigned mirror is numbered

Measuring Height w Similar Triangles

Angle-Angle Triangle Similarity ShortcutAA(A) proves similarity – screaming helps!

5 m 2 m

Scale factor = 5m/2m = 2.5

5 m 2 m

Scale factor = 5m/2m = 2.5

Scale factor x small height = 2.5 x 1.5m = 3.75m big height

Method #2aThe Held-Out Ruler Method

Measuring Height or Distancewith Similar Triangles

Method #2bThe Better Held-Out Ruler Method

Method #3The Shadow Method

3-day projectIndirectly Measure Heights of Tall

Objects on Campuswith Similar Triangles

• Each partner records data & calculations.• Measure each height two ways - mirror &

ruler methods:1. Gold ball on flagpole outside2. Ceiling in commons3. Bottom edge of U.S. flag in commons4. Water tower (tricky; can’t measure

directly to base…use similar triangles, twice?)

Mirror

Friday 1/9 – Measure Heights of Tall Objects with Similar Triangles

You are graded on clean up & returning materials intact…

Place mirror, meter stick, little ruler on desk to be checked in.

• HW due tomorrow 11.3 #1-10.• Quiz tomorrow, sections 1-3.• May use your notes.• May NOT use cell phone

calculator for scale factors.

14.0 cm tower shadow on Google Maps, from base

4.0 cm flag pole shadow on Google Maps

Indirect measurement, using shadows

1) Calculate scale factor from flag & tower shadows

2) Scale factor x known flag height = tower height

Unit Review

Partners Warm-Up – 10 min• On whiteboard, graph polygon:

A (3, 4), B (2, -4), C (-5, -4)• Draw dilations centered on the origin

with scale factors of 2:1 final:initial (2x bigger) and 1:2 (1/2x smaller). Move vertices along rays…• Label A’, A’’…

Solo – 10 min• On graph paper in your notebook,

graph polygon:A (-3, -4), B (-2, 4), C (5, -4)

• With center of dilation on point D (3, 4),•Dilate 2:1 final:initial (2x)•1:2 (1/2x)• -2:1 (-2x) Hint: Draw -1 dilation

Solo – 10 min

• On graph paper in your notebook, graph polygon:• Complete steps for question 2,

p. 617.

Solo – 10 min

• On graph paper in your notebook, graph polygon:• Complete steps for question 1,

p. 617.• Discuss with group, write down your

conclusion and be ready to explain your conclusions.

Partners Warm-Up – 10 min

• On graph paper in your notebook, graph polygon:

A (-3, -4), B (-2, 4), C (5, -4)• Draw 2:1 (2x) and 1:2 (1/2x) and -2:1

(-2x) dilations centered on point O (3, 4). Move vertices along rays…

Agenda

• QUIZ – Yes: notes & calculator. No: cell phones.

Review for Test

1. In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5). Tape in graph paper if needed.

2. Draw rays from origin through each vertex.3. Dilate above shape with scale factors of 0.5

and 2.4. Write as complete sentences:“In a dilation, _________ angles ___________.”“We can control the scale factor by _____________________________________.”

Review for Test

1. In notebook, graph (3, 2) (4, -1) (-3, 3) (-2, -5). Tape in graph paper if needed.

2. Draw rays from origin through each vertex.3. Dilate above shape with scale factors of 0.5

and 2.4. Write as complete sentences:“In a dilation, corresponding angles remain congruent.”“We can control the scale factor by increasing the

distance between center-of-dilation & vertices by that factor.”

Dilation using ‘ray method’1. Draw a circle with your compass.2. Place a point outside the circle for

the center of dilation.3. Use the ‘ray method’ to dilate the

circle 2x bigger. Don’t need a graph.4. Dilate the original circle ¾ of original

size, using only compass & straight edge.

Dilations Centered on Same PointDilations work for all polygons…

Center of dilation can be inside shapes.

Center of dilation can be outside shapes.

Dilations Centered on Same PointScale factor can be negative…

What is the scale factor in this dilation?

1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (5, 0) (6, -3) (-1, 1) (0, -7).

2. Write as complete sentence:“These polygons are: congruent/similar/neither (may be more than one) because __________ _____________________.3. “If similar, the scale factor is ____________.”

1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (1.5, 2) (2, -1) (-1.5, 3) (-1, -5).

2. Write as complete sentence:“These polygons are: congruent/similar/neither (chose one) because _____________________.3. “If similar, the scale factor is ____________.”

1. Compare polygon (3, 2) (4, -1) (-3, 3) (-2, -5) and polygon (1.5,1) (2,-0.5) (-1.5,1.5) (-1,-2.5).

2. Write as complete sentence:“These polygons are: congruent/similar/neither (chose one) because _____________________.3. “If similar, the scale factor is ____________.”

Friday 1/16

• 30 min• No notes• No cell phone• Yes calculator

• 10/10 = A+• 9/10 = A-• 7/10 = B• 5/10 = C• 3/10 = D

TEST – Similarity & Dilations

A level topics 11.4 & 11.5

• Verify geometric properties of dilations

WarmUp – 5 min• Draw a square 2 cm on each side.• Draw a square 6 cm on each side.• Draw dashed lines to show how many little

squares fit in the big square.• Calculate the area of each and write in each

square with units.• Fill in this Conjecture:

“If corresponding sides of two similar polygons compare in a ratio of m/n, then their areas compare in the ratio of ______________.”

Proportional Volume Conjecture• Draw a cube 2 cm on each side.• Draw a bigger cube 6 cm on each side.• Draw dashed lines to show one little cube in

the corner of the big cube.• Calculate the volume of each cube.• Fill in this Conjecture:

“If corresponding edges (or radii or heights) of two similar solids compare in a ratio of m/n, then their volumes compare in the ratio of ______________.”

For shapes with a scale factor of 2,how do these ‘scale up’?

• Perimeter?• Area?

For solids with scale factor of 2,how do these ‘scale up’?

• Surface Area?• Volume?

For shapes with a scale factor of 3,how do these ‘scale up’?

• Perimeter?• Area?

For solids with scale factor of 3,how do these ‘scale up’?

• Surface Area?• Volume?

Worksheet to Complete

Lesson 11.5 – Proportions with Area and Volume

1) Which two HW Q’s would you most like to see?

2) Working with your partner, readpp. 599-602

‘Why Do Elephants Have Big Ears?’

Discuss, then write your answers for Q’s 1-15. Show Mr. Sidman.

1. You will need your compass.2. Place a point in the center of the notebook page.3. Draw three rays ‘radiating’ out from this point.4. Randomly place one additional point on each ray.

Place them at the edges of the paper. Connect them to form a BIG triangle.

5. Place a second point at half the distance along each ray as the first. This makes a similar triangle with a scale factor of 0.5 compared to the first.

6. Bisect one corresponding side of each triangle.7. Construct one corresponding median for each

triangle. Find the ratio of big-to-little medians.8. Bisect a corresponding angle in each. Find the ratio

of the big-to-little angle bisector segments.

Conclusion

• Corresponding (matching) dimensions of similar triangles all have the same scale factor (ratio):

little side = little median = little angle bisector big side big median angle bisectorlittle altitude = little midsegment = little perpendicular bisector

big altitude big midsegment big perpendicular bisector

1. You will need your compass.2. Place a point in the center of the notebook page.3. Draw three rays ‘radiating’ out from this point.4. Randomly place one additional point on each ray.

Place them at the edges of the paper. Connect them to form a BIG triangle.

5. Place a second point at half the distance along each ray as the first. This makes a similar triangle with a scale factor of 0.5 compared to the first.

6. Find the ratio of the small-to-big perimeters.7. For one corresponding angle in each triangle, drop a

perpendicular bisector to the opposite side.8. Use this altitude (height) to find the ratio of areas.

WarmUp – 5 min

1. Write a step-by-step proof that∆LMN ∆EMO.̴

2. What is length y?

1. Draw two rays forming an acute angle.2. On one ray, use a ruler to mark off lengths 8 cm and

then an additional 10 cm from vertex. Label these segments.

3. On the other ray, mark off segments 12 cm and an additional 15cm.

4. Connect points to make a little triangle inside the big.5. Are these triangles similar?6. Calculate the ratio 8cm to 10cm. 12cm to 15cm.

What do you notice?7. What else do you notice about these triangles?

Ch. 11 - Similarity Class Notes. What Is Similarity?

Documents