Download Math Institute April 2010 Most Missed Questions: Applying Basic

Connecting Ratios, Proportions,
Percents and Similar Triangles
Presenters:
Angie Kaldro & Cheryl Sanders
April 2012
Objectives:
• To share strategies on teaching the interrelationships of
ratios, proportions and similar triangles.
• Teachers will learn how ratios and proportions are used
to solve similar triangle word problems as well as using
the percent triangle as a viable manipulative.
• Impartation of large and small group activities that will
include invaluable handouts and scaffolding activities.
Golden Ratio
Everything Is Connected To The Golden Ratio Chris
MintZ Barf
http://www.youtube.com/watch?feature=player_detailpage&v
=U2bAlIK4KkE&list=PLFD81A492266D0372
Ratios
??USAGES??
Ratios
Relationship between two numbers
• Cost per Unit
• Probability
• Scale/size
• Rate
Ratios
Relationship between two numbers
• Cost per Unit
• Probability
• Scale/size
• Rate
--Comparison of 2 quantities by division-
Unit Rates
• Kilo per Hour
• Beats per minute
• Points per game
• Dollars per hour
Ratios and Proportions
• Solving Word Problems
Proportion Box (PB)
• Effective with more challenged students
• Complete activity using Proportion Box
Solving Percents by Proportions
•
Set up percent questions using proportions
•
Base is to Part as 100% is to the %
Percents
Every percent problem has three possible unknowns,
or variables:
a. the percent
b. the part
c. the base
In order to solve any percent problem, you must
be able to identify these variables.
The Percent Triangle
Part
Percent
Base
Δ
Sample GED Question
A department store advertises a clearance sale that
offers “Take an additional 40% off the sale price.”
A coat that was originally $75 is on sale for $50.
What is the clearance price?
$20
$25
$30
$40
$50
Similar Triangles
Definition:
• Triangles are similar if they have the same shape, but
can be different sizes.
• They are still similar even if one is rotated or one is a
mirror image of the other.
Properties of Similar Triangles
1.
Corresponding angles are the
congruent (same measure).
2.
Corresponding sides are all in the same
proportion.
Properties of Similar Triangles
1.
Corresponding angles are the congruent (same
measure).
So in the figure above, the angle B=E, C=F, and A=D
Properties of Similar Triangles
2. Corresponding sides are all in the same
proportion. AB = BC =AC
DE
EF
DF
Properties of Similar Triangles
•
DE is twice the length of AB. Therefore, the other
pairs of sides are also in that proportion. EF is
twice BC and DF is twice AC.
•
Formally, in two similar triangles ABC and DEF:
AB= BC =AC
DE EF DF
SO WHAT IS X?
Properties of Similar Triangles
Activity
So if: AB= BC
DE EF
Then:
10 = 6
x 12
6x = 10 x 12
6x = 120
x = 20
Properties of SIMILAR TRIANGLES
Rotation: One triangle can be rotated, but as long as
they are the same shape, the triangles are still
similar.
Properties of SIMILAR TRIANGLES
Reflection: One triangle can be the mirror image of
the other, but as long as they are the same
shape, the triangles are still similar. It can be
reflected in any direction, up, down, left,
right.
How To Tell if Triangles Are Similar
Any triangle is defined by six measures (three sides,
three angles). But you don’t need to know all of them to
show that two triangles are similar. Various groups of
three will do. Triangles are similar if:
1. AAA (angle, angle, angle): All three pairs of
corresponding angles are the same.
2. SSS in same proportion (side, side, side): All three
pairs of corresponding sides are in the same proportion.
3. SAS (side, angle, side): Two pairs of sides in the same
proportion and the included angle equal.
Similar Triangles Can Have Shared
Parts
•
Two triangles can be similar, even if they share
some elements.
•
In the figure below, the larger triangle ABC is
similar to the smaller one A’BC’.
•
They are similar on the basis of AAA, since the
corresponding angles in each triangle are same.
Similar Triangles Can Have Shared
Parts Activity
Problem 1:In the triangle ABC shown below, A'C' is parallel
to AC. Find the length y of BC' and the length x of A'A.
Similar Triangles Can Have Shared
Parts Activity
Solution to Problem 1:
BA is a transversal that intersects the two parallel lines A'C' and AC, hence the
corresponding angles BA'C' and BAC are congruent. BC is also a transversal to the
two parallel lines A'C' and AC and therefore angles BC'A' and BCA are congruent.
These two triangles have two congruent angles are therefore similar and the lengths
of their sides are proportional. Let us separate the two triangles as shown below.
Similar Triangles Can Have Shared
Parts Activity
Similar Triangles Can Have Shared
Parts Activity
We now use the proportionality of the lengths of the side to write equations that help in
solving for x and y.
(30 + x)/30 = 22 / 14 = (y + 15) / y
An equation in x may be written as follows.
(30 + x) = 22
30
14
Solve the above for x.
420 + 14 x = 660
x = 17.1 (rounded to one decimal place).
An equation in y may be written as follows.
22 = (y + 15)
14
y
Solve the above for y to obtain.
y = 26.25
Are These Two Triangles Similar?:
Connecting Ratios and Proportions
Step 1. Check to see if the figures appear to have the
same shape.
Step 2. Find the ratio of the hypotenuses.
Step 3. Find the ratio of the shorter legs.
Step 4. Find the ratio of the longer legs.
Set up the proportions:
Are These Two Triangles Similar?:
Connecting Ratios and Proportions
Set up the Ratios:
Hypotenuse: 7
3
Shorter Legs: 5
2
Longer Legs: 2
1
7/3 is not equal to 5/2 is not
equal to 1.
Since the ratios are not equal, the triangles are not
similar.
QUESTIONS
???????????????
Copyright © 2010 Texas Education Agency
Copyright © Notice. The materials are copyrighted © and trademarked ™ as the property of the Texas Education Agency (TEA) and may not be
reproduced without the express written permission of TEA, except under the following conditions:
Texas public school districts, charter schools, and Education Service Centers may reproduce and use copies of the Materials and Related Materials
for the districts’ and schools’ educational use without obtaining permission from TEA.
Residents of the state of Texas may reproduce and use copies of the Materials and Related Materials for individual personal use only without
obtaining written permission of TEA.
Any portion reproduced must be reproduced in its entirety and remain unedited, unaltered and unchanged in any way.
No monetary charge can be made for the reproduced materials or any document containing them; however, a reasonable charge to cover only the
cost of reproduction and distribution may be charged.
Private entities or persons located in Texas that are not Texas public school districts, Texas Education Service Centers, or Texas charter schools or
any entity, whether public or private, educational or non-educational, located outside the state of Texas MUST obtain written
approval from TEA and will be required to enter into a license agreement that may involve the payment of a licensing fee or a
royalty.
For information contact
Richard Jarrell
Office of Copyrights, Trademarks, License Agreements, and Royalties
Texas Education Agency
1701 N. Congress Ave.
Austin, TX 78701-1494
(512) 463-9270 or (512) 936-6060