Math 2
G.SRT.4
Assessment Title: Pythagorean Theorem by Similarity
Unit 6: Similarity
Learning Target:
 Students will prove the Pythagorean Theorem using similar triangles.
Students completed several proofs of the Pythagorean Theorem in grade 8. Pair students and direct
them to complete a proof of the Pythagorean Theorem. They may use any method, or you may direct
them to complete the proof in a specific manner. Walk around the room assisting students as needed.
Provide time for students to discuss their proofs with the class. An example of an algebraic proof they
might remember from grade 8 is below.
An Algebraic Proof: In the picture to the left, all
triangles are right and congruent and have side
lengths of 𝑎, 𝑏, and 𝑐 as noted. Notice that the area
of the large outer square is equal to the area of the
four right triangles plus the area of the small inner
square. Let’s set-up an equation representing these
figures and see what we get.
Big Square = Triangles + Small Square
𝑎
𝑏
𝑐
Statement
1. The area of the big square is
Reason Why It’s True
1.
2. The area of one triangle is
2.
3. The area of four triangles is
3.
4. The area of the little square is
4.
The equation representing the area of the big
square equal to the area of the triangles plus the
area of the small square is
5. Simplifying this equation gives us
5.
After completing and discussing the familiar proof(s) of the Pythagorean Theorem, present students
with the following picture of a right triangle with altitude to the hypotenuse. Discuss the three triangles
in the picture. This discussion should lead to the three triangles being similar. Now ask students to use
the similar triangles to prove the Pythagorean Theorem. Facilitate as necessary.
A
Given: ABC with right angle B
D
BD  AC
Prove: (AB)2 + (BC)2 = (AC)2
B
C
Students completed several proofs of the Pythagorean Theorem in grade 8. Pair students and direct
them to complete a proof of the Pythagorean Theorem. They may use any method, or you may direct
them to complete the proof in a specific manner. Walk around the room assisting students as needed.
Provide time for students to discuss their proofs with the class. An example of an algebraic proof they
might remember from grade 8 is below.
An Algebraic Proof: In the picture to the left, all
triangles are right and congruent and have side
lengths of 𝑎, 𝑏, and 𝑐 as noted. Notice that the area
of the large outer square is equal to the area of the
four right triangles plus the area of the small inner
square. Let’s set-up an equation representing these
figures and see what we get.
Big Square = Triangles + Small Square
𝑎
𝑏
𝑐
Statement
1. The area of the big square is
𝑐2
2. The area of one triangle is
1
𝑎𝑏
2
3. The area of four triangles is
1
4 ∗ 𝑎𝑏 = 2𝑎𝑏
2
4. The area of the little square is
(𝑎 − 𝑏)(𝑎 − 𝑏) = 𝑎(𝑎 − 𝑏) − 𝑏(𝑎 − 𝑏)
= 𝑎2 − 𝑎𝑏 − 𝑏𝑎 + 𝑏 2
= 𝑎2 − 𝑎𝑏 − 𝑎𝑏 + 𝑏 2
= 𝑎2 − 2𝑎𝑏 + 𝑏 2
The equation representing the area of the big
square equal to the area of the triangles plus the
area of the small square is
𝑐 2 = 2𝑎𝑏 + 𝑎2 − 2𝑎𝑏 + 𝑏 2
5. Simplifying this equation gives us
𝑐 2 = 𝑎2 + 𝑏 2
Reason Why It’s True
1. Area of a square is 𝑠 2
1
2. Area of a triangle is 2 𝑏ℎ
3. Multiply by four
4. Distributive Property (distribute the first
parentheses)
Distributive Property again
Commutative Property changing −𝑏𝑎 to – 𝑎𝑏
Combining like terms where – 𝑎𝑏 − 𝑎𝑏 = −2𝑎𝑏
5.
Combine like terms where 2𝑎𝑏 − 2𝑎𝑏 = 0
After completing and discussing the familiar proof(s) of the Pythagorean Theorem, present students
with the following picture of a right triangle with altitude to the hypotenuse. Discuss the three triangles
in the picture. This discussion should lead to the three triangles being similar. Now ask students to use
the similar triangles to prove the Pythagorean Theorem. Facilitate as necessary.
A
Given: ABC with right angle B
D
BD  AC
Prove: (AB)2 + (BC)2 = (AC)2
B
C
A
B
A
B
C
C
D
D
Statements
1. ABC with right angle B
BD  AC
2. BDC and ADB are right angles
3. BAC  DAB
ACB  BCD
4. ABC  BDC  ADB
Reasons
1. Given
5. BAC DAB
ACB BCD
AB AC
BC AC
6.
and


AD AB
DC BC
7. (AB)2  AD(AC ) and (BC )2  DC (A C )
8. (AB)2  (BC )2  AD(AC )  DC (AC )
9. (AB)2  (BC )2  AC (AD  DC )
10. AD + DC = AC
11. (AB)2  (BC )2  AC (AC )
12. (AB)2  (BC )2  (AC )2
5. Angle Angle Similarity
2. definition of perpendicular
3. reflexive property of congruence
4. All right angles are congruent.
6. definition of similarity
7. cross product property
8. substitution
9. distributive property
10. Segment Addition Postulate
11. substitution
12. simplify
B