Transcript Chapter 7.1 & 7.2 Notes: The Pythagorean Theorem and its

Chapter 7.1 & 7.2 Notes: The
Pythagorean Theorem and its
Converse
Goal: To use the Pythagorean Theorem and its
Converse.
Right Triangles:
• In a right triangle, the side opposite the right angle is
the longest side, called the hypotenuse. The other
two sides are the legs of a right triangle.
• Theorem 7.1 Pythagorean Theorem:
In a right triangle, the sum of the squares of the
lengths of the legs is equal to the square of the
length of the hypotenuse.
a 2 + b 2 = c2
• Find the value of x. Leave your answer in simplest
radical form.
Ex.1:
Ex.2:
Ex.3: A 16-foot ladder rests against the side of the
house, and the base of the ladder is 4 feet away.
Approximately how high above the ground is the top
of the ladder?
• When the lengths of the sides of a right triangle are
integers, the integers form a Pythagorean Triple.
• Common Pythagorean Triples and Some of Their
Multiples:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
3x, 4x, 5x
5x, 12x, 13x
8x, 15x, 17x
7x, 24x, 25x
6, 8, 10
10, 24, 26
16, 30, 34
14, 48, 50
9, 12, 15
15, 36, 39
24, 45, 51
21, 72, 75
30, 40, 50
50, 120, 130
80, 150, 170
70, 240, 250
Ex.4: Find the area of the isosceles triangle with side
lengths 10 meters, 13 meters, and 13 meters.
• Find the value of x. Leave your answer in simplified
radical form.
Ex.5:
Ex.6:
• Theorem 7.2 Converse of the Pythagorean
Theorem:
If the square of the length of the longest side of a
triangle is equal to the sum of the squares of the
lengths of the other two sides, then the triangle is a
right triangle.
If c2 = a2 + b2, then ∆ABC is a right triangle.
• Tell whether a triangle with the given side lengths is
a right triangle.
Ex.7: 5, 6,
Ex.8: 10, 11, 14
61
Ex.9:
8
4
4 3
• The Converse of the Pythagorean Theorem is used
to determine if a triangle is a right triangle, acute
triangle, or obtuse triangle.
– If c2 = a2 + b2, then the triangle is a right triangle.
– If c2 > a2 + b2, then the triangle is an obtuse
triangle.
– If c2 < a2 + b2, then the triangle is an acute
triangle.
• Determine if the side lengths form a triangle. If so,
classify the triangle as acute, right, or obtuse.
Ex.10: 15, 20, and 36
Ex.11: 6, 11, and 14
Ex.12: 8, 10, and 12
Ex.13: 4.3, 5.2, and 6.1