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

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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 partc. the base

In order to solve any percent problem, you must be able to identify these variables.

The Percent Triangle Δ Part

Percent Base

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 TrianglesActivity

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 3Shorter Legs: 5

2Longer Legs: 2 17/3 is not equal to 5/2 is not equal to 1.

Since the ratios are not equal, the triangles are not similar.

Since the ratios are not equal, the triangles are not similar.

QUESTIONS

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