12.4 similar triangles and similar...
TRANSCRIPT
12.4 Similar Triangles and Similar Figures
Definition of Similarity
Definition∆ABC is similar to ∆DEF, denoted as ∆ABC ∼ ∆DEF, if and onlyif the corresponding angles are congruent and the corresponding sidesare proportional.
A•
46
D•
23
E•2
•F
B•4
•C
ABDE
=BCEF
=ACDF
Question
Explain the following:Are all isosceles triangles similar?
Ways to Prove Similarity of Triangles
TheoremSSS Similarity for TrianglesIf the lengths of corresponding sides of two triangles areproportional, then the triangles are similar.
Ways to Prove Similarity of Triangles
TheoremSAS Similarity for TrianglesIf two sides are proportional to the corresponding sides and theincluded angles are congruent, then the triangles are similar.
Ways to Prove Similarity of Triangles
TheoremAA Triangle SimilarityIf two angles in one triangle are congruent to the correspondingangles in another triangle, then the triangles are similar.
Example
Triangle SimilarityExplain why ∆DBE ∼ ∆ABC. AA
What is the length of BE?
412
=x
x + 98x = 36
x =92
Example
Triangle SimilarityExplain why ∆DBE ∼ ∆ABC. AA
What is the length of BE?
412
=x
x + 98x = 36
x =92
Example
Triangle SimilarityExplain why ∆ABC ∼ ∆ADB. AA
Find the value of x.
Solution
x3
=x + 4
66x = 3(x + 4)
6x = 3x + 12
3x = 12
x = 4
Notice now that the length of the side AD is twice the length of AB,giving us a ratio of 1
2 for the measures of the sides in ∆ABCcompared to the corresponding sides of ∆ADE.
Solution
x3
=x + 4
66x = 3(x + 4)
6x = 3x + 12
3x = 12
x = 4
Notice now that the length of the side AD is twice the length of AB,giving us a ratio of 1
2 for the measures of the sides in ∆ABCcompared to the corresponding sides of ∆ADE.
Theorem
TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.
TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.
TheoremIf a parallel line cuts off congruent segments on one tranversal, thenthey cut off congruent segments on any transversal.
Theorem
TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.
TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.
TheoremIf a parallel line cuts off congruent segments on one tranversal, thenthey cut off congruent segments on any transversal.
Theorem
TheoremIf a line parallel to one side of a triangle intersects the other sidesthen it divides those sides into proportional segments.
TheoremIf a line divides two sides of a triangle into proportional segments,then the line is parallel to the third side.
TheoremIf a parallel line cuts off congruent segments on one tranversal, thenthey cut off congruent segments on any transversal.
Midpoints
DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.
TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.
TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.
Midpoints
DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.
TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.
TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.
Midpoints
DefinitionThe midsegment is the segment connecting the midpoint of adjacentsides of a triangle or quadrilateral.
TheoremThe Midpoint TheoremThe midsegment joining the midpoint of two sides of a triangle isparallel to and is half as long as the third side.
TheoremIf a line bisects one side of a triangle and is parallel to a second sidethen it bisects the third side and therefore is a midsegment.
Centroid
DefinitionThe median of a triangle is the segment joining a vertex and themidpoint of the opposite side.
DefinitionThe centroid is the point of concurrency of the three medians of atriangle.
Centroid
DefinitionThe median of a triangle is the segment joining a vertex and themidpoint of the opposite side.
DefinitionThe centroid is the point of concurrency of the three medians of atriangle.
Example
Triangle SimilarityExplain why ∆abc ∼ ∆fde
Example
Suppose Susie wants to figure out how tall a tree is. She stands 10 feetfrom a tree and notices that she casts a 6 foot shadow and that hershadow ends in the exact same place as that of the tree. If Susie is 5feet tall, how tall is the tree?
Example
Suppose Susie wants to figure out how tall a tree is. She stands 10 feetfrom a tree and notices that she casts a 6 foot shadow and that hershadow ends in the exact same place as that of the tree. If Susie is 5feet tall, how tall is the tree?
Solution
First, we have to justify that the triangles are similar. What are thetwo triangles?
56
=x
166x = 80
x =403
So, the tree is 13 13 feet tall.
Solution
First, we have to justify that the triangles are similar. What are thetwo triangles?
56
=x
166x = 80
x =403
So, the tree is 13 13 feet tall.
Example
SimilarityFind the value of z.
610
=4z
z =203
Example
SimilarityFind the value of z.
610
=4z
z =203
Example
More SimilarityJustify why these triangles are similar and then find the value of x andy.
12x
=1620
=20y
So, x = 15 and y = 25.
Example
More SimilarityJustify why these triangles are similar and then find the value of x andy.
12x
=1620
=20y
So, x = 15 and y = 25.
Example
More SimilarityJustify why these triangles are similar and then find the value of x andy.
12x
=1620
=20y
So, x = 15 and y = 25.
Similarity and Other Polygons
DefinitionAny two polygons with the same number of sides are similar if andonly if the corresponding angles are congruent and the correspondingsides are proportional.
Same idea without the ‘named’ theorems and postulates.
Similarity and Other Polygons
DefinitionAny two polygons with the same number of sides are similar if andonly if the corresponding angles are congruent and the correspondingsides are proportional.
Same idea without the ‘named’ theorems and postulates.
Example
Similarity
Suppose you wanted to make a copy of a document at 18 of the original
size, but you made a mistake and made a copy of the original at 25 of
the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1
8 of the original size. What ratio should you use to do this?
We think of this as 18 is the part we want and 2
5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives
1825
=x
1005
16=
x100
16x = 500
x = 31.25
Example
Similarity
Suppose you wanted to make a copy of a document at 18 of the original
size, but you made a mistake and made a copy of the original at 25 of
the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1
8 of the original size. What ratio should you use to do this?
We think of this as 18 is the part we want and 2
5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives
1825
=x
100
516
=x
10016x = 500
x = 31.25
Example
Similarity
Suppose you wanted to make a copy of a document at 18 of the original
size, but you made a mistake and made a copy of the original at 25 of
the original size. You are stubborn, so instead of starting at over, youwant to use the copy you made and reduce it to make the final productbe 1
8 of the original size. What ratio should you use to do this?
We think of this as 18 is the part we want and 2
5 is the whole, since thatis what we are working with now. But we want to know what part ofthe original whole this corresponds to. This gives
1825
=x
1005
16=
x100
16x = 500
x = 31.25
Similarity
ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?
We haveABDE
=70100
,GHDE
=30100
ABDEGHDE
=70
10030
100
abgh
=7030
So, the ratio between the first and third triangles is 73 .
Similarity
ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?
We haveABDE
=70100
,GHDE
=30100
ABDEGHDE
=70
10030
100
abgh
=7030
So, the ratio between the first and third triangles is 73 .
Similarity
ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?
We haveABDE
=70100
,GHDE
=30100
ABDEGHDE
=70
10030
100
abgh
=7030
So, the ratio between the first and third triangles is 73 .
Similarity
ExampleSuppose we have ∆ABC, ∆DEF, and ∆GHI such that ∆ABC is 70%of ∆DEF and ∆GHI is 30% of ∆DEF. What is the ratio between∆ABC and ∆GHI?
We haveABDE
=70100
,GHDE
=30100
ABDEGHDE
=70
10030
100
abgh
=7030
So, the ratio between the first and third triangles is 73 .
Similarity and Slope
Do you remember the formula for slope? How about the phrase weuse when working with slope?
Slope
m =riserun
=y2 − y1
x2 − x1
How does this relate to similar triangles?
Similarity and Slope
Do you remember the formula for slope? How about the phrase weuse when working with slope?
Slope
m =riserun
=y2 − y1
x2 − x1
How does this relate to similar triangles?
Similarity and Slope
Do you remember the formula for slope? How about the phrase weuse when working with slope?
Slope
m =riserun
=y2 − y1
x2 − x1
How does this relate to similar triangles?