Introduction Our project covers sections 8.4, 8.5, and 8.6. These sections discuss the side...

Introduction

Our project covers sections 8.4, 8.5, and 8.6. These sections discuss the side splitting theorem,

indirect measurement and additional similarity theorems, and area and volume ratios. The

material covered in our presentation utilizes the skills that we have been learning in previous

chapters throughout the semester. These skills are quite useful when trying to figure out which

parts of different triangles are proportional to each other, and they provide other ways to determine if

multiple triangles are similar and why.

This side of a triangular shaped building is a perfect demonstration

of the side splitting theorem because it contains multiple lines that are parallel to one side of the triangle. These lines all divide the

triangle proportionally.

Side Splitting Theorem

• A line parallel to one side of the triangle divides the other two sides proportionally.

• This theorem applies to any segment that is parallel to one side of a triangle.

• www.nhvweb.net/.../13.3%20The%20Side-Splitting%20Theorem.doc shows an example of how the side splitting theorem can be used to find the length of a missing side with proportions.

Different Proportions Found From The Side Splitting Theorem

• UL == UR UL == UR UL == LL

• --------------- --------------- -------------

• LL LR WL WR UR LR

• LL == LR

• -------------

• WL WR

Two Transversal Proportionality Corollary

• Three or more parallel lines divide two intersecting transversals proportionally.

A B

C D

E F

This diagram contains several triangles. It demonstrates the

Two Transversal Proportionality Corollary in multiple ways

because there are a total of 9 parallel lines dividing 3 different

pairs of intersecting transversals.

•All of the proportions previously mentioned are relevant here as

well.

Review Questions

• What is the Side Splitting Theorem and how is it used in Geometry?

• How can the Side Splitting Theorem be helpful in understanding which parts of a triangle are proportional to one another?

8.4-8.6

By: Forest Schwartz and Max reenhouse

• Finding the length of X, given two triangles are similar can be very useful for a variety of jobs. A military engineer needs to build across a river, therefore he needs to know how long it is.

Using Geometry for uses

C

X

72 M

40 m

30 m

A

B E

D

• Using the Diagram above, we can state those two triangles using AA, as we learned in the previous sections. First say vertical angles, which proves angle BCA = ECD. Now we can say angle B = 90 and C= 90 degrees because they are both right angles. Use the transitive property to declare they are congruent.

Proportions

• Now that we have said the triangles are similar, we can set up a proportion.

•

EC

BC=

DE

AB

5

12=

40

X

x 96=405

12=X

•Therefore X=96

Proportional Medians Theorem

If two triangles are similar, then their corresponding medians have the same ratio as their corresponding sides.

Proportional Altitudes Theorem

IF the two triangles are similar, then their corresponding altitudes have the same ratio as their corresponding sides.

Proportional Angle Bisectors Theorem

• If two triangles are similar, then their corresponding angle bisectors have the same ratio as the corresponding sides.

Proportional Segments Theorem

• An angle bisector of a triangle divides the opposite side into two segments that have the same ratio as the other two sides.

20

10

5

x

Area

• The area of a triangle or square is the same ratio of the sides, just squared.

• 3/1 is the ratio of the sides

• 3/1 is the ratio of the areas

Sample Problems

• If the ratio of the preimage to the image is 1:3 in a square, then what is its ratio of the area.

• Ratio of preimage of triagle to the image is 2:4, so what is its ratio of the areas.