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19.2 Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. In a right triangle, the two legs form the right angle, while the side opposite this angle is called the hypotenuse. This theorem states that if 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse, then the relationship can be expressed as (a^2 + b^2 = c^2). Through various examples, including special right triangles like 45-45-90 and 30-60-90 triangles, we explore how to find missing side lengths using this theorem.

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19.2 Pythagorean Theorem

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  1. 19.2 Pythagorean Theorem

  2. A right triangle is a triangle that has a right (90 degree) angle. The 2 sides that form the right angle are called the legs of the triangle and the side opposite the right angle is called the hypotenuse. hypotenuse leg leg leg hypotenuse leg

  3. The Pythagorean Theorem relates the sides of any right triangle. If a and b represent the legs and c the hypotenuse, then

  4. Find the length of the hypotenuse: c 24 dm 10 dm

  5. Find the length of the hypotenuse: 8.00 cm 3.90 cm c

  6. Find the length of the missing side 25.0 m 18.0 m b

  7. Find the length of the missing side 42,600 ft 37,800 ft b

  8. 15 m 10 m • How long must a guy wire be to reach from the top of a 15-m telephone pole to a point on the ground 10 m from the foot of the pole? • Use the Pythagorean theorem. c

  9. There are two special right triangles that we will further explore. They are called a 45, 45, 90 right triangle and a 30, 60 ,90 right triangle. The numbers refer to the measure of the angles in degrees.

  10. The 45, 45, 90 triangle is an isosceles triangle which means the two legs have equal measure. leg leg 45 45 hypotenuse Since the legs are equal, by the Pyth.Theorem, the hypotenuse has length = leg

  11. Find the hypotenuse of an isosceles triangle that has equal sides of 2 m. 2 m 2 m ? Hypotenuse has length =

  12. If the hypotenuse of a 45, 45, 90 triangle is 5 cm long, how long is each leg. ? 5 cm Since hyp = leg

  13. An equilateral triangle has 3 equal sides and 3 equal angles that each measures 60 degrees. The height bisects both the angle and opposite side making a 30, 60, 90 triangle. 30 ½ side or ½ hypot 60 60

  14. Applying the Pythagorean Th. to this 30, 60, 90 triangles gives the following relations: 30 2x, side opposite right angle or longest side other leg 60 x, side opposite 30 or short leg

  15. Find the missing measures. 30 Hyp. = 2(short leg) = 2(7 in) = 14 in ? Other leg = short leg ? 60 7 in

  16. Find the missing measures. 60 ? Other leg = short leg ? 30 8 cm Hyp. = 2(short leg) = 2( 4.6in) = 9.2 in

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