Chapter 3 Similarity
Day 1
3.1 Similar Polygons
3.2 Applications of Similar Polygons
Have balls and rubber bands at the ready!
3.3 Pythagorean Theorem
Calculator with inverse trig functions (“” key)
Day 2
3.3 Laws of Sine and Cosine, continued
Calculator with inverse trig functions
3.4 Area and Perimeter of Similar Figures
3.5 Similarity for More General Figures
3.1 Similar Polygons
What is a good definition for similar polygons…what do you remember about similarity?
Now let’s check our definition of similar against the one on page 94…defn 3.1.1
And let’s look at that notation at the bottom of the page closely: see also the naming convention at the very bottom
Is similarity an equivalence relation? What does that mean?
Let’s look at the similarity ACTIVITY now.
Page 97 – see the set diagram. What is it saying?
All polygon pairs are congruent?
Each pair of congruent polygons are also similar?
Let’s discuss set containment
3.2 Applications of Similar Triangles
Proportional work with heights:
Page 99 using shadows to find heights….sun convention
See the proportion equation on page 100
See the picture on page 101. Check out the equation and how to solve it.
Page 102 – VERY IMPORTANT Theorem 3.2.1 (AA Similarity)
If two triangles have two pairs of corresponding angles that are congruent, then the triangles are similar.
Note: this only works in Euclidean Geometry! Let’s look at a counterexample from Spherical Geometry!
SAS Similarity – CAUTION with this
SSS Similarity
Do the “nested triangles” similarity exercise in the ACTIVITIES now.
Finding similar triangles with a right triangle and an altitude
(defn, p. 104)
What are the three triangles?
Let’s set up the three pairs to check together. Then go to the ACTIVITIES pages and do the work to find them similar.
Note that the AA similarity theorm is about TRIANGLES page 105 top
And NOT about any other polygons!
3.3 The Pythagorean Theorem
There are MANY proofs of the Pythagorean Theorem. In 1947 The Pythagorean Proposition was printed by Professor Elisha Scott Loomis. The book is a collection of 367 proofs of the Pythagorean Theorem and has been republished by NCTM in 1968.
We’ll look at only one or two of them.
Now let’s redo these into similar triangles with the right angle in the lower left. Rename
AD = C1 and DB = C2. So C = C1 + C2. Label sides “a” and “b” across from these angles.
Note that a/c = c2/a and c/b = b/c1. Let’s go from there.
This is on pages 107 and 108 in the book.
Converse of the Pythagorean Theorem
PT: If triangle ABC is a triangle with a right angle at C, then .
Page 7: original; converse.
State the converse of the Pythagorean Theorem and TPT, compare!
Proof of converse: given a triangle where the equation is true…show that C is a right triangle in the original triangle.
Construct a second triangle with the same leg lengths and C’ is right.
Now, use the equation with the second coordinates a’, b’, and c’ via the PT.
Substitute with the hypothesis equation on the right a and b for the primed…substitute c on the right…c = c’…SSS congruence. Now C is right by CPCF.
Trigonometry – page 110
Given a right triangle, let’s map out the 6 trig functions by relationship.
Trigonometric Ratios - know by heart!
Given a right triangle, the following trigonometric ratios are defined:
“SOHCAHTOA”
Let’s get the Big Three sine, cosine, and tangent. Let’s look at “cos inverse” on our calculators and get the angle measures.
Note the one in the book: 5 – 12 – 13, bottom page 110
Page 110 + enrichment
Trigonometry is based on right triangles. Let’s review the 6 trig functions on a right triangle.
Right triangles and trigonometric ratios
Given a right triangle,
The Pythagorean Theorem holds.
Apply it to the triangle on the right:
TPT can be applied to situations when the unknown
is something other than the hypotenuse. Solve
for x:
Let’s use our calculators to get the measure of angle B and angle A
And a discussion of the other angles: b & q
Note: b + q = 90° COMPLEMENTS not supplements
The angles b and q are ACUTE angles of the right triangle. A is the right angle.
There can only be one right angle or obtuse angle in a triangle in EG!
What is a formula for the measure of angle b?
(n.b. It’s not independent of Ðq !)
Two Special Triangles:
Note that 30° - 60° - 90° go together in a right triangle.
Isosceles right triangles are the second special triangles.
Mnemonic: 30° - 60° - 90° :: small, medium, big :: x, , 2x
(note: )
Check those sides? Are they right?
sin 30° sin 60°
inverse sine of the sine…
cos 30° cos 60°
inverse cosine of the cosine…
tan 30° tan 60°
If you have an isosceles triangle with side length 6 cm, how can a 30-60-90 triangle help you with the length of the altitude?
What is the height of the triangle?
and 45° - 45° - 90° go together in an isosceles right triangle.
Check those side lengths! Are they right?
Note the hypotenuse is the longest side in any right triangle
sin 45° cos 45°
inverse sine inverse cosine
tan 45°
Given an isosceles right triangle with a hypotenuse of 5cm, what is the leg length?
ACTIVITIES: Supplemental Angle Conundrum!
MORE VOCABULARY:
The angles 30°, 45°, and 60° are all called “reference angles”.
0°, 90°, 180°, 270°, and 360° are all called “quadrantal angles”.
Angles ON the axes, not IN a particular quadrant
Ok, now before the famous chart. Let’s look at TANGENT.
Why is tangent = sine/cosine?
You have to know ALL of the special angle material by heart.
BUT, happily, I have a nice little mnemonic device for you below
Ms. Leigh’s Famous Chart
Count off left to right starting with 0.
Count back right to left starting with 0.
Square root and divide by 2.
angle in deg / 0° / 30° / 45° / 60° / 90°angle in rad
[for later]
sine
cosine
tangent
Textbook pages 112 – 114
Law of Sines and Law of Cosines
All the trigonometry information was based on having a right triangle as the starting point. It turns out that some trigonometric facts can be extended to arbitrary triangles.
Let’s look at a triangle:
This is called an obtuse triangle, because it is not a right triangle and angle C is greater than 90°. What can we find out about this triangle – well, the area is ½ AC(CB)sin 110°. True, but we only know one side!
We’ll use the very handy, but problematic Law of Sines to solve this problem
If you have ANY triangle the following equation holds:
A problem: find the lengths of the sides!
Measure of angle C is 110°.
Example: completely describe this triangle in exact measures, no rounding.
Note what you know is AAS, just like the preceding triangle!
Why do I focus on cosine inverse?
Sin (110) = Sin(70) =
Cos(110) = Cos(70) =
There’s only one wrinkle to the Law of Sines…if you know SSA you can’t be sure of your answer because the sine function has the same values for it’s Q1 and Q2 supplementary angles. This is called the Ambiguous Case for the Law of Sines.
Let’s look at an example of this
In triangle MNO, the measure of angle M is 30°, the length of NO is r, and the length of MO is , find all possible measures for angle N.
What’s the answer if the measure of angle M is 150°? Does the MATH change?
The key to this problem is that these two angles add to 180°; they are supplements.. When you use arcsin in your calculator, you will only get the Quadrant 1 angle (why?), you have to supply the Quadrant 2 angle on your own if, indeed, you need it.
Example:
Given triangle ABC, find the measure of angle A, given that B = 32°, a = 42 and b = 30. “a” is the side across from A.
Let’s sketch the triangle and go to work using the Law of Sines.
How many answers might there be?
(hint using a calculator )
Summary:
Triangle ABC have a, b, measure angle b…what is measure of angle a?
Sin A = −.3 Sin A = 1.7 no solution
Sin A = .87 and the triangle has an obtuse angle one solution
and both B and C are acute one or 2 solutions
Now on to the law of cosines…another formula to KNOW BY HEART.
It turns out that if you know 3 sides of a triangle you can get all the angle measures…and if you know SAS you can get the third side. You use this formula:
There is no ambiguity when using this formula the answer from your calculator is the only answer.
How many solutions: ONE
Suppose we have the following scenario. What is BC?
Or we have
What is the cos(A)?
How will we find the measure of angle A?
Example
Given triangle ABC, the measure of angle A is 45°, the length of AB is 5, and the length of AC is , what is the length of BC?
Do the Law of Cosines and the Law of Sines ACTIVITIES now
Section 3.4 Area and Perimeter of Similar Figures
Page 116, text
Let’s discuss the constant of proportionality and its role in finding the area and perimeter of similar figures.
Given a 3 – 4 – 5 right triangle and one that has a k = 0.5. Let’s sketch these and calculate both area and perimeter of each
Now let’s analyze where the 1/2 fits in from a formula sense and make a rule to fit.
Let’s pick an equilateral triangle with side length 3. What is the perimeter and area?
Then use a constant of proportionality k = 2. What is the perimeter and area?
What is the rule for perimeter of similar figures?
What is the rule for area of similar figures?
WHY do these rules work?
Do the area and perimeter exercise in the ACTIVITIES pages now
One way to make similarity transforms happen is to give instructions for changing the graph in a sort of functional notation.
Your original shape is f(x, y) where x and y are coordinates. If you want k to be 2, you’d write f(2x, 2y).
Let’s look at this with a triangle (0, 0), (0, 1) and (2, 0). Graph this. What is the area? Now apply the instruction f(2x, 2y). Graph it and what is the area? What is the k?
We will be doing the exercise at the top of page 120
Graph Hat A (0, 4), (0, 1), (6, 1), (4, 2), (4, 4), (3, 5), (1, 5), (0, 4)
This is the basic hat. Go back to it for each additional hat and analyze the transforms.
What is the area enclosed by the hat? Use Pick’s Theorem:
Now transform it to Hat B: ( x + 2, y + 2) what changed?
Points:
What is the area of the hat?
Hat C: (x + 3, y – 1)
Points
What is the area of the hat?
What is this movement called?
Hat D: (2x, y + 2)
Points
What is the area of the hat?
How do we describe these changes?
Hat E: (2x, 3y)
Points
What is the area of the hat?
What changed and how do we describe it?
Hat F: (½ x, ½ y)
Points:
WITHOUT calculating: what is the area of this hat?
Let’s summarize what the instructions do to a set of points:
F(2x – 1, 3y + 2) What are the instructions?
Similarity transformations – a transformation that takes a pair of points A, B to a pair A’,B’ such that the original distance between them is multiplied by the given scale factor. Similarity transforms are FUNCTIONS.
a, b, c, d, h, k will be GIVEN.
Similarity transforms that create similar polygons are (kx, ky) where k is the scale factor and not equal to zero. If k is negative it takes the object “through” the vanishing point. We call this a similarity transform with the reflection about the origin.
3.5 upping the ante on transformations.
Sculptors and architects design small models of structures in advance of the real work. Suppose a sculptor designs a model with a volume of 5 cubic inches and his scale factor to the real thing is 15. What will the increase in the volume be?
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