Transcript Right Triangles and the Pythagorean Theorem

RIGHT
TRIANGLES AND
THE
PYTHAGOREAN
THEOREM
Lilly Weston
Curriculum 2085
University of Louisiana at Monroe
Table of
Contents
• What’s a right triangle and what makes it special?
• A right triangle versus an isosceles
• What does the Pythagorean Theorem mean?
• What are a, b, and c?
• When would you use this theorem?
• Problems
– Is this a right triangle?
– Missing side’s length?
– Word problem
– Will these measurements create a right triangle?
• Who created this theorem?
• Review part one
• Review part two
What’s a
right
triangle and
what makes
it special?
Things all triangles share:
• Three sides
• Three angles
• All the angles, when added together, equal 180°
Only right triangles have:
• A 90° angle
• A hypotenuse
• The three sides are given specific names
– a, b, and c
• The Pythagorean theorem can ONLY be used on these
– a2 + b2 = c2
A right
triangle
versus an
isosceles
Right triangle
22
12
16
Another triangle (isosceles)
15
15
25
What does
the
Pythagorean
theorem
mean?
• The sum of the two areas of the two squares (a and b)
equals the area of the square of the hypotenuse (c).
– a2 + b2 = c2
• Visual proof
• More info about this is at Khan Academy
What are a, b,
and c?
• Legs – a and b
• create the 90° angle
• Hypotenuse – c
• this side will always be
on the opposite side of
the 90° angle
• the longest of the three
sides
c
a
90°
b
When would
you use this
theorem?
• You would use this if you’re unsure if a triangle is a
right, if you have two sides of the right triangle and
you’re trying to find the length of the third side, or for
certain word problems.
• Example: If a = 5 and b = 9, what will c equal?
• Example:
7
b=?
12
What are
the steps?
Find c if a = 12 and b = 14. Round to the
nearest tenth.
• Step 1: Write the equation down.
– a2 + b2 = c2
• Step 2: Plug in the information to the
correct variable.
– (12)2 + (14)2 = c2
*For this problem, we
had to use the square
root on both sides of
the equation.
• Step 3: Solve – this isn’t always the same
for each problem.*
Before you put down
– 144 + 196 = c2
your answer, make
– 340 = c2
– √(340) = √(c2)
sure to look at how it
– 18.4 = c
should be rounded!
Is this a
right
triangle?
a2 + b2 = c2
10
8
6
(8)2 + (6)2 = (10)2
64 + 36 = 100
*For this problem, we
100 = 100
didn’t have to use the
Is this a right triangle?
square root. All we
needed to do was square
everything and add on
both sides of the equal
sign.
YES*
What is the
length of the
missing side?
Round to the
nearest tenth.
3
7
1. a2 + b2 = c2
2. (7)2 + (3)2 = c2
3. 49 + 9 = c2
54 = c2
√(54) = √(c2)
7.6 = c
c
Word Problem
1. A 30 foot ladder is leaning against a
wall. The bottom of the ladder is 6 feet
from the base of the wall. How tall is
the wall?
30 feet
1. a2 + b2 = c2
2. a2 + (6)2 = (30)2
a=?
3. a2 + 36 = 900
-36
0
-36
864
a2 = 864
√(a2) = √(864)
a = 29.4 feet
6 feet
Will these
measurements
create a right
triangle?
• a = 6.4, b = 12, c = 12.2
1. a2 + b2 = c2
2. (6.4)2 + (12)2 = (12.2)2
3. 40.96 + 144 = 148.84
184.96 ≠ 148.84
NO
Who created
this theorem?
•
•
From the Greek island Samos
•
•
Pythagoras (570 - 495 B.C.)
Died in Metapontum, Italy
Because there is little written work of his, some
are unsure that he contributed to mathematics
at all.
•
Influenced Plato and all of Western Philosophy
Review – part one
1.
T or F? You can use this equation on any type of triangle?
FALSE; it can only be used on right triangles
2. T or F? Pythagoras was born in Italy and died on a Greek island.
FALSE; he was born on a Greek island (Samos) and died in Italy (Metapontum)
3.
Could these create a right triangle? a = 12.5, b = 14, c = 11. Do not work out the problem.
NO; the hypotenuse, c, must be the longest of the three sides
4. What is the length of the missing side(to the nearest tenth)?
a2 + b2 = c2
a2 + (18) 2 = (28)2
28
a2 + 324 = 784
a
-324 -324
0
460
a2 = 460
√(a2) = √(460)
18
a = 21.4
Review –
part two
1.
Where, on this triangle, does each letter belong?
c
a
b
2.
3.
What are the difference(s) between a right triangle
and an equilateral triangle?
1.
R: one 90° angle + 2 other angles, side lengths are not
necessarily the same length
2.
E: three 60° angles, all three sides are the same length
What is true of the hypotenuse, c, no matter what the
situation?
1.
It is always the longest side.