Download Pythagorean Theorem 1

Pythagorean Theorem
Name_______________________
Day 1
Period______Date_____________
If an altitude AC is dropped from the
right angle in a right triangle, three triangles
are formed.
What’s true about these 3 triangles and why?
Remembering that corresponding sides of similar
triangles are proportional, three extended proportions can be written. Write these below.
The geometric mean between a and b is a number x so that
a x
= . There are 3 geometric mean
x b
statements in the extended proportions above. Write those geometric means below.
Use this information to solve the following problems:
1. If CD = 9 and BC = 4, find AC.__________
Then find AD. __________
2. If BC = 2 and BD = 8, find AB.__________
Then find AC.___________
3. If CD = 6 and BC = 3, find AD.__________
Then find AB.___________
In each of the above problems, you could use the Pythagorean Theorem to work the last part.
That’s because one proof of the Pythagorean Theorem uses this geometric mean.
Given: h is the altitude to the hypotenuse of
right triangle ABC.
Prove: a2 + b2 = c2
1. h is the altitude to the hypotenuse of
right triangle ABC.
1. Given
2. ADC and BDC are right angles
2. Altitudes form right angles.
3. ACD  BDC  ACB
3. All right angles are congruent.
4. A  A and B  B
4.
5. ∆ABC ~ ∆ACD and ∆ABC ~∆CBD
5.
6.
y a
x b
= and
=
b c
a c
7. b2 = x∙c and a2 = y∙c
6.
7.
2
2
8.
2
2
9. a + b = c∙(x + y)
9.
10. x + y = c
10.
11. a2 + b2 = c∙c = c2
11.
8. a + b = x∙c + y∙c
Actually there are 364 proofs of the Pythagorean Theorem.
Find another one of them and bring it in tomorrow!
If ∆ PQR is a right triangle with right angle Q, solve each of the following problems:
1. If p = 5 and r = 7, find q._______________
2. If p = 8 and r = 6, find q._______________
3. If p = 3 and q = 8, find r. _______________
4. If r = 5 and q = 13, find p.______________
Problems 2 and 4 are special because the right triangles in those problems had integer values for
all 3 sides. The numbers {6, 8, 10} and {5, 12, 13} are called Pythagorean triples because they are
integer values so that a2 + b2 = c2. Pythagorean triples and similarity relationships are often found
on standardized tests.
There are many more Pythagorean triples. Write down any odd number. ________
Square that odd number and divide it by 2. _________. Round this number up _______
and round this number down. ________
The 3 integers will form a Pythagorean triple. Prove
that below.
{3, 4, 5}, {5, 12, 13}, {7, 24, 25} and {8, 15, 17} are the most common Pythagorean triples. Knowing
these and similar triangle properties can make finding sides of right triangles easier.
Consider the following: Two legs of a right triangle are 30 7 and 40 7 . What is the
hypotenuse?
Solution: The common factor of these 2 numbers is 10 7 , and if we factor out that common
factor, we get that legs of a similar triangle are 3 and 4. The hypotenuse of a triangle with legs of
3 and 4 is 5, so the hypotenuse of our triangle is 5(10 7 ) = 50 7 .
Try solving these:
1. Two legs of a right triangle are 15 and 36. Find the hypotenuse.___________
2. The hypotenuse of a right triangle is 100. If one leg has length 28, find the other
leg._______
The distance formula is based on the Pythagorean Theorem too.
Plot (2, 5) and (-1, 1).
Draw a right triangle on this grid with
these points as the endpoints of the hypotenuse.
Show how the Pythagorean Theorem is used to find
the distance between these points.
Pythagorean Theorem
Name_______________________
Day 2
Period______Date_____________
Review:
In the figure to the right, b = 60 and c = 156.
Find each of the following:
a = ___________
d = ______________
e = ___________
f = ______________
As you saw yesterday, some right triangles have all integral side lengths these side lengths are
called Pythagorean Triples. Other triangles are special because of their angles. Consider the
following situation:
AM is the altitude of EQUILATERAL triangle ABC. Construct that triangle below and altitude AM.
What is mABM?________, mAMB________
Let the side of your triangle be any even
Number. ___________
Why is MC = MB?
What is MC?__________
Use the Pythagorean Theorem to find AM.
B
C
∆ABM is called a 30°-60°-90° triangle. Consider the equilateral triangle below with side length 2s.
The altitude will create 2 30°-60°-90° triangles.
Clearly, the shorter leg of the 30°-60°-90°
triangle will be half the length of the hypotenuse.
Show work to find the length of the altitude,
which is the longer leg of the 30°-60°-90° triangle.
The longer leg of a 30°-60°-90°is
3 times the length of the shorter leg.
Try these:
1. If the shorter leg of a 30°-60°-90° triangle is 17, find the hypotenuse:_________ and the
longer leg.___________
2. If the hypotenuse of a 30°-60°-90° triangle is 100, find the shorter leg: ________ and the
longer leg.___________
3. If the longer leg of a 30°-60°-90° triangle is 6, find the shorter leg:_________ and the
hypotenuse._____________
Pythagorean Theorem
Name_______________________
Day 3
Period______Date_____________
Review:
In the figure to the right, b = 60 and mA = 60.
Find each of the following:
d = ___________
f = ______________
e = ___________
a = ______________
c = ___________
State why each of the following pairs of triangles are congruent:
____________
_______________
_________________
____________
ΔABC  ΔEDC
ΔABE  ΔACD
ΔADT  ΔEDT
ΔPRO  ΔPST
P
R
O
T
S
You know triangles cannot be proven congruent using “SSA” because that theorem does not exist.
HOWEVER, consider the two RIGHT triangles ΔABC and ΔXZY.
If C and Y are right angles, legs, AC = ZZ = 15 and
hypotenuses AB  XZ = 17, that is an SSA situation.
But – aren’t these triangles congruent?
What makes this situation different is that in a right triangle, if we know the lengths of 2 sides,
we know the length of the third side. That means the SSA case works IF the triangle is a right
triangle. However, since it is a special case, we call this the HL theorem, which means “hypotenuseleg”.
And it reads:
If there exists a correspondence between the vertices of two right triangles such that
the hypotenuse and a leg of one triangle are congruent to the corresponding parts of
the other triangle, then the two right triangles are congruent. (HL)
Consider the following proof:
D
Given: AB  BC, DB is an altitude
Prove: DB is an angle bisector
A
B
C
1. AB  BC, DB is an altitude
1. Given
2. DB  AC
2.___________________________
3. ABD and CBD are right angles.
3. ___________________________
4. BD  BD
4. ___________________________
5. ΔABD  ΔCBD
5. ___________________________
6. BDA  BDC
6. ___________________________
7. DB is an angle bisector.
7. ___________________________
More fun with right triangles!!
You will be assigned an integer. Write that integer here:__________
Show work to find the length of the diagonal of a square with the side length written above.
You used a right triangle to solve this problem. What are the angles in this triangle?
Compare notes with your partner. Can you generalize the situation?
This triangle is called a 45-45-90 triangle and this is another special right triangle because of
its angles.
So, in conclusion:
In any 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the
shorter leg and the length of the longer leg is
3 times the length of the shorter leg.
and
In any 45°-45°-90° triangle, the legs are congruent and the hypotenuse is
length of a leg.
2 times the