Download Unit 5 Proving the Pythagorean Theorem using Similar Triangle

Unit 5 Proving the Pythagorean Theorem using Similar Triangle
Proving the Pythagorean Theorem using Similar Triangle
Unit 5:
April 19, 2016
Given: A right triangle with an altitude(height) drawn from the right angle to the hypotenuse.
C
Prove: a2 + b2 = c2
Objective:Students will be able to prove the pythagorean theorem using similar triangles. We’ll partition ABC into 3 similar right triangles and use the equal ratios of their corresponding parts in our proof.
b
A
a
D
x
B
y
Draw a perpendicular to the hypotenuse AB through the vertex of the right angle at C.
That perpendicular intersects AB at a point we’ll call D.
c
C
Now we have 3 similar right triangles:
ABC, the big one we started with, b
containing A and B,
ACD, the little one
that contains A,and A
BCD, the other little one that contains B.
We know all 3 triangles are similar by AA:
two pairs of equal corresponding angles make similar triangles.
C
a
x
B
D y
c
How do we know all 3 triangles are similar?
Each triangle has a right angle, and all right angles are equal. And each triangle contains either A
or or both
a
b
x
B
D y
c
Similar triangles means that the ratios of their corresponding parts will be equal.
Similar triangles means that the ratios of their corresponding parts will be equal.
C
C
C
A
x
a
b
b
D
A
x
D y
C
b
a
B
D y
B
A
x
c
1.Make proportions(ration) using all corresponding sides.
C
a
b
D A
C
x
D y
c
1.Make proportions(ration) using all corresponding sides.
2. Substitute your proportion with lower case letters .
2. Substitute your proportion with lower case letters .
3. Cross multiply .
3. Cross multiply .
a
B
D y
B
Unit 5 Proving the Pythagorean Theorem using Similar Triangle
April 19, 2016
From picture what is y + x equal to?
2. add both equations write down your answer.
Substitute.
C
a2 = cy
a2 + b2 = c(y+x)
b2 = cx
a
b
a2 + b2 = cy + cx
3. factor the right side of the equation
a2 + b2 = c(x+y)
A
x
D y
B
c
1. Cross Multiply both proportions
1
4
Example
Use the pythagorean theorem to solve the unknown. Where necessary, round you answer correct to one decimal place. B
A
5
Example
In shop class, you make a table. The sides of the table measure 36" and 18". If the diagonal of the table measures 43", is the table “square”? (In construction, the term "square” just means the table has right angles at the corners.)
Example
A ladder is leaning against the side of a 10m house. If the base of the ladder is 3m away from the house, how tall is the ladder? Draw a diagram and show all work.
Attachments
Proving the Pythagorean Theorem using Similar Triangles.docx
Assig Unit 5 Proving Pythagorean thm .docx