Transcript 7-2 PPT Pythagorean Theorem

Section 11-2 The Pythagorean Theorem
SPI 32A: apply the Pythagorean Theorem to real life problem illustrated by a
diagram
Objectives:
• Solve problems using the Pythagorean Theorem
Right Angle:
• angle that forms 90°
Hypotenuse:
• in a right triangle, it is the side opposite the 90° angle
Leg:
• each of the sides forming the right triangle
Pythagorean Theorem:
• describes the relationship of the lengths of sides of a
right triangle.
The Pythagorean Theorem
In any right triangle, the sum of the squares of the lengths
of the legs is equal to the square of the length of the
hypotenuse. . . a2 + b2 = c2
c (hypotenuse)
This is a right triangle
a
90°
(leg)
b
(leg)
a and b are the legs of the right triangle
This is NOT a right triangle
c is the hypotenuse and is
ALWAYS the longest segment
A Pythagorean Triple
A set of nonzero whole numbers a, b, and c that satisfy the
Pythagorean Theorem formula.
If you multiply each number in a Pythagorean Triple by a
whole number, the resulting numbers will also form a Triple.
a b  c
2
2
2
Common Pythagorean Triples
3, 4, 5
5, 12, 13
8, 15, 17
32  4 2  5 2
9  16  25
52  12 2  132
25  144  169
82  15 2  17 2
64  225  289
The Pythagorean Theorem
What is the length of the
hypotenuse of this triangle?
Step 1. Write the formula for the
Pythagorean Theorem.
a2  b2  c2
Step 2. Substitute in known values.
82  152  c 2
64  225  c 2
Step 3. Solve for the unknown variable.
289  c
17  c
2
Real-world Pythagorean Theorem
A television screen measures approximately
15.5 in. high and 19.5 in. wide. A television is
advertised by giving the approximate length
of the diagonal of its screen. How should this
television be advertised?
Draw and label a diagram to model the problem.
15.5
c
19.5
Solve for c, using the Pythagorean Theorem.
a2  b2  c2
15.52  19.52  c 2
240.25  380.25  c 2
620.5  c 2
The television should be advertised as a 25”.
24.9  c
Real-world Pythagorean Theorem
A toy fire truck is near a toy building on a table such
that the Base of the ladder is 13 cm from the building.
The ladder is extended 28 cm to the building. How high
above the table is the top of the ladder?
Draw a diagram to model
the problem.
Solve using the
Pythagorean Theorem
a2  b2  c2
132  b 2  282
169  b 2  784
b 2  615
b  24.8
The truck is approximately (24.8 + 9)
33.8 cm above the table.
Converse of the Pythagorean Theorem
If a triangle has sides of length a and b, and a2 + b2 = c2, then
the triangle is a right triangle with hypotenuse of length c.
Remember, c is the longest side in a right triangle.
Determine whether the given lengths are sides of a right
triangle.
a. 5 in., 5 in., and 7 in.
a2  b2  c2
Not a right triangle
52  52  7 2
25  25  49
b. 10 cm, 24 cm, and 26 cm a 2  b 2  c 2
10 2  24 2  26 2
This is a right triangle
100  576  676
Physics and the Pythagorean Theorem
If two forces pull at right angles to each other, the resultant
force is represented as the diagonal of a rectangle, as
shown in the diagram. The diagonal forms a right triangle
with two of the perpendicular sides of the rectangle.
For a 50–lb force and a 120–lb force, the
resultant force is 130 lb. Are the forces pulling
at right angles to each other?
a2  b2  c2
50  120  130
2
2
2
2500  14400  16,900
Yes, the forces are
pulling at right angles
to each other.
Real-world Connection and the Pythagorean Theorem
A baseball diamond is a square with 90-ft sides. Home
plate and second base are at opposite vertices of the
square. About how far is home plate from second base?
Use the information to draw a baseball diamond.
a2 + b2 = c2
902 + 902 = c2
8100 + 8100 = c2
Use the Pythagorean Theorem.
Substitute 90 for a and for b.
Simplify.
16,200 = c2
c = 16,200
c
Take the square root.
127.27922 Use a calculator.
The distance to home plate from second base is about 127 ft.