Mod 2 Part II Notes

NAME _____________________________________
COMMON CORE GEOMETRY
Module 2 Part II
Setting up proportions with similar triangles
And Simplifying Radicals
DATE
12/11
PAGE
2-3
HOMEWORK
Homework Worksheet
20-22
TOPIC
Lesson 1: Similar triangles and proportions
Start Lesson 2
LESSON 2: Applying the Triangle Side Splitter
Theorem
LESSON 3: Applying the triangle side splitter
theorem continued
Lesson 4: Ratios of Sides, Perimeters, and Areas
QUIZ
Lesson 5: The Angle Bisector Theorem
Lesson 6: Special Relationships Within Right
Triangles—Dividing into Two Similar SubTriangles
Lesson 7: Special Relationships Within Right
Triangles- Another useful proportion
QUIZ
LESSON 8: A side note- what to do if we get a
radical?
Lesson 9: More operations with radicals
12/12
4-5
12/15
6-7
12/16
12/17
8-10
11-12
12/18
13-15
12/19
16-17
12/22
18-19
12/23
1/5/15
23-24
LESSON 10: Adding and Subtracting Radicals
Homework Worksheet
1/6
25-26
LESSON 11: Putting It all Together
Homework Worksheet
QUIZ
Review
Review
Test
Finish Review Packet
1/7
1/8
1/9
Homework Worksheet
Homework Worksheet
Homework Worksheet
Homework Worksheet
Homework Worksheet
Homework Worksheet
No Homework
No Homework
Study
No Homework
1
Lesson 1: Similar triangles and proportions
Prove triangles are similar:
a.) In the diagram below DE is parallel to AB, mark your picture accordingly:
C
D
E
A
B
b.) Fill in the appropriate givens and what you are trying to proveGiven:
Prove:
c.) Proof:
d.) Draw the similar triangles separately and label:
D.)Now that we know that the triangles are similar, Let’s fill in the following proportions:
2
EXERCISES: In 1-3 find x given that ̅̅̅̅
̅̅̅̅
1.
2.
3.
4. A vertical pole, 15 feet high, casts a shadow 12 feet long. At the same time, a nearby tree casts a shadow 40
feet long. What is the height of the tree?
5. Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet
above the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the
cliff with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from the
cliff, to the nearest tenth, is Catarina’s boat?
5.5ft
10ft
35ft
3
LESSON 2: Applying the Triangle Side Splitter Theorem
Side Splitter Theorem:
A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.
Using this Theorem, answer the following questions:
1. If ̅̅̅̅
̅̅̅̅ , ̅̅̅̅
, ̅̅̅̅
2. If ̅̅̅̅
̅̅̅̅ , ̅̅̅̅
, ̅̅̅̅
, and ̅̅̅̅
, and ̅̅̅̅
, what is ̅̅̅̅ ?
, what is ̅̅̅̅ ?
4
4.
5.
6. In the diagram pictured, a large flag pole stands outside of an office building. Josh realizes that when he
looks up from the ground, 60m away from the flagpole, that the top of the flagpole and the top of the building
line up. If the flagpole is 35m tall, and Josh is 170m from the building, how tall is the building to the nearest
tenth?
5
LESSON 3: Applying the triangle side splitter theorem continued
Opening Exercise
In the two triangles pictured below, ̅̅̅̅
̅̅̅̅ and ̅̅̅̅
̅ . Find the measure of x in both triangles
What is the relationship between the triangles ABC and FGH?
Since the two triangles share a common side, look what happens when we push them together:
Now we have three parallel lines cut by transversals, is the transversal on the left in proportion to the
transversal on the right?
6
THEOREM: If 3 or more lines are cut by 2 transversals, then the segments of the transversals are in
proportion.
Exercises
In exercises 1 and 2, find the value of x. Lines that appear to be parallel are in fact parallel
1.)
2.)
3.) In the diagram below, ̅̅̅̅ ̅̅̅̅ ̅̅̅̅, AB=20, CD=8, FD=12 and AE:EC=1:3. If the perimeter of the
trapezoid ABCD is 64, find AE and EC.
7
Lesson 4: Ratios of Sides, Perimeters, and Areas
Exercise
Given ABCA’B’C’ pictured to the right:
a. Find the lengths of the missing sides.
b. Find the perimeters of the triangles.
c. Find the areas of the triangles.
Area formula:
d. What is the ratio of the sides of the triangles?
e. What is the ratio of the perimeters of the triangles?
f. What is the ratio of the areas of the triangles?
RULES:

The ratio of the perimeters is _____________________________________________________

The ratio of the areas is _________________________________________________________
8
Example 2
Let’s test the hypothesis we made in Exercise 1.
Given the similar triangles pictured to the right, find:
a.) The ratio of the sides
b.) The ratio of the perimeters
c.) The ratio of the areas
Given the similar rectangles pictured below, find:
a.) The ratio of the sides
b.) The ratio of the perimeters
c.) The ratio of the areas
9
Examples:
1.) When two figures are similar and the ratio of their sides is a:b, then:
The ratio of their perimeters is:
The ratio of their areas is:
2.) Two triangles are similar. The sides of the smaller triangle are 6,4,8. If the shortest side of the larger
triangle is 6, find the length of the longest side.
3.) The sides of a triangle are 8, 5, and 7. If the longest side of a similar triangle measures 24, find the
perimeter of the larger triangle.
4.) Find the ratio of the lengths of a pair of corresponding sides in two similar polygons if the ratio of the
areas is 4:25.
10
Lesson 5: The Angle Bisector Theorem
The angle bisector theorem states:
In ABC, if the angle bisector of A
meets side BC at point D, then
𝐵𝐷
𝐶𝐷
𝐵𝐴
𝐶𝐴
Exercises:
1.) In ABC pictured below, AD is the angle bisector of A. If CD=6, CA=8 and AB=12, find BD.
2.) In ABC pictured below, AD is the angle bisector of A. If CD=9. CA=12 and AB=16. Find BD.
11
3.) The sides of ABC pictured below are 10.5, 16.5 and 9. An angle bisector meets the side length of 9. Find
the lengths of x and y.
12
Lesson 6: Special Relationships Within Right Triangles—Dividing into Two Similar Sub-Triangles
Opening Exercise
Use the diagram to complete parts (a)–(c).
a. Are the triangles shown similar? Explain.
b. Determine the unknown lengths of the triangles using Pythagorean Theorem.
Example 1
In ABC pictured to the right, B is a right angle and BC is the altitude.
DEFINE: Altitude: ____________________________________________________________________
a. How many triangles do you see in the figure? Draw them:
the altitude ̅̅̅̅ divides the right triangle into two sub-triangles,
b. In
Since
Is
?
Is
?
and
can we conclude that
13
and
? Explain.
Example 2
Consider the right triangle
below.
8
4
a. Altitude ̅̅̅̅ is drawn from vertex
and the segment ̅̅̅̅ as
x
to the line containing ̅̅̅̅ Segment ̅̅̅̅
, segment ̅̅̅̅
,
b. Draw the two similar triangles ADB and BDC and label the sides appropriately.
c.
Set a proportion to find the value of
d. We should notice that the altitude of the whole triangle is the long leg in the small triangle and is the
short leg in the medium triangle. Therefore we can really use a short cut when we see a diagram like
this:
14
EXAMPLES:
1.)
3.) Given triangle
2.)
with altitude ̅ , find
16
,
.
36
15
Lesson 7: Special Relationships Within Right Triangles- Another useful proportion
Consider the right triangle
below (same as we worked with yesterday).
x
4
20
a. Altitude ̅̅̅̅ is drawn from vertex
and the segment ̅̅̅̅ as
to the line containing ̅̅̅̅ Segment ̅̅̅̅
, segment ̅̅̅̅
,
b. Draw the two similar triangles ADB and ABC and label the sides appropriately.
c.
Set a proportion to find the value of
d. We should notice that the hypotenuse of the small triangle is the leg in the large triangle and they
hypotenuse in the large triangle is the leg in the short triangle. Therefore we can really use a short cut
when we see a diagram like this:
16
EXAMPLES:
1.)
2.)
what is the length of
?
What is the length of
3.) In the diagram below, the length of the legs
and
respectively. Altitude
is drawn to the hypotenuse of
a centimeter?
17
?
of right triangle ABC are 6 cm and 8 cm,
. What is the length of
to the nearest tenth of
LESSON 8: A side note- what to do if we get a radical?
Opening Exercise
Solve for x:
x
6
8
Consider our answer, what can we do with it? If we round we are not using the most accurate answer
so we want to leave it in simplest radical form. (we’ll come back and do this later)
Perfect Squares- ________________________________________________________________________
Identify some perfect squares below:
Factors:
Perfect square
Factors:
12
xx
22
x2x2
33
X3x3
42
X4x4
52
X5x5
62
X6x6
72
X7x7
Perfect square
82
92
*make a note about this:
102
112
18
Write the square roots of the following:
1.
2. √
√
4. √
3. √
5. √
Simplifying Non-Perfect squares:
√𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑟𝑒
√𝐿𝑎𝑟𝑔𝑒𝑠𝑡 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 ∗ √𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟
Simplify:
6. √
7. √
8. √
9. √
10. √
11.) go back and simplify
your answer from the
opening exercise.
19
Lesson 9: More operations with radicals
1.
Complete parts (a) through (c).
a. Compare the value of √
to the value of √
√
b. Make a conjecture about the validity of the following statement. For nonnegative real numbers
,√
√ √ . Explain.
c.
2.
Does your conjecture hold true for
and
and
?
Complete parts (a) through (c).
a. Compare the value of √
to the value of
√
√
.
b. Make a conjecture about the validity of the following statement. For nonnegative real numbers
, when
c.
,√
√
√
. Explain.
Does your conjecture hold true for
and
20
?
and
Exercises 3–12
Simplify each expression as much as possible and rationalize denominators when applicable.
3.
√
√
5.
√
√
7.
√
9.
√
10.
√
12.
11.
√
4.
√
6.
√
8.
21
√
√
√
√
13.
Find the area of the figure below:
14.
Calculate the area of the triangle:
22
LESSON 10: Adding and Subtracting Radicals
1.)
a. Calculate the perimeter of the triangle below:
b. Calculate the perimeter of the triangle:
√
√
√
Since the radicals are not the same we need to do some work before we can add.
*Simplify each side then add.
23
SIMPLIFY
2.)
√
4.) √
√
√
.
3.) √
√
√
√
5.)
Find the Perimeters
6.)
7.)
24
−√
LESSON 11: Putting It all Together
Opening Exercise
In the diagram below of right triangle ABC, an altitude is drawn to the hypotenuse
.
Recall the proportions that we set up back in lessons 6 and 7 and fill in the appropriate proportions below:
𝒙
𝒄
𝒄
𝒛
𝒃
𝒂
1.) Triangle ABC shown below is a right triangle with altitude
If
and
drawn to the hypotenuse
.
Write all answers in simplest radical form:
a.) what is the length of AD?
C.) what is the length of AC?
b.) What is the length of AB?
D.) What is the area of triangle ABC?
25
.
2.) Given the diagram below, find:
4
16
a.) Length of AB
b.) Length of AD
c.) Length of AC
D.) Perimeter of triangle ABC.
E.) Express the area of triangle ABC in simplest form:
26

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