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Questions and Answers
What is the Pythagorean theorem and how is it related to special right triangles?
What is the Pythagorean theorem and how is it related to special right triangles?
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In special right triangles, such as the 45-45-90 and 30-60-90 triangles, the ratios of the sides can be used to easily calculate the length of the hypotenuse or other sides.
A ladder is leaning against a wall at a 60 degree angle. The base of the ladder is 8 feet away from the wall. How long is the ladder?
A ladder is leaning against a wall at a 60 degree angle. The base of the ladder is 8 feet away from the wall. How long is the ladder?
The ladder is 16 feet long.
A flagpole casts a shadow that is 24 feet long. At the same time, a person standing nearby casts a shadow that is 6 feet long. If the person is 5 feet tall, how tall is the flagpole?
A flagpole casts a shadow that is 24 feet long. At the same time, a person standing nearby casts a shadow that is 6 feet long. If the person is 5 feet tall, how tall is the flagpole?
The flagpole is 20 feet tall.
What is the length of the hypotenuse of a 45-45-90 triangle if each of the legs has a length of 5√2?
What is the length of the hypotenuse of a 45-45-90 triangle if each of the legs has a length of 5√2?
A 30-60-90 triangle has a shorter leg of length 3. What is the length of its hypotenuse?
A 30-60-90 triangle has a shorter leg of length 3. What is the length of its hypotenuse?
A right triangle has one leg of length 12 and a hypotenuse of length 13. What is the length of the other leg?
A right triangle has one leg of length 12 and a hypotenuse of length 13. What is the length of the other leg?
Flashcards
Pythagorean Theorem
Pythagorean Theorem
In a right triangle, a² + b² = c², where 'c' is the hypotenuse. Useful for finding unknown side lengths.
Special Right Triangles
Special Right Triangles
In 45-45-90 triangles, legs are equal, Hypotenuse = leg * √2. In 30-60-90 triangles, sides are in ratio 1:√3:2.
Hypotenuse of 45-45-90 Triangle
Hypotenuse of 45-45-90 Triangle
It measures 10 units
Hypotenuse of 30-60-90 Triangle
Hypotenuse of 30-60-90 Triangle
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Missing Leg of Right Triangle
Missing Leg of Right Triangle
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Study Notes
Pythagorean Theorem and Special Right Triangles
- The Pythagorean theorem is a fundamental concept in geometry that relates to special right triangles.
- In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Problem-Solving Applications
- In a ladder problem, if the ladder is leaning against a wall at a 60-degree angle and the base is 8 feet away from the wall, the length of the ladder can be calculated using the Pythagorean theorem.
- In a shadow problem, if a flagpole casts a shadow that is 24 feet long and a person standing nearby casts a shadow that is 6 feet long, the height of the flagpole can be calculated using similar triangles.
45-45-90 and 30-60-90 Triangles
- In a 45-45-90 triangle, the length of the hypotenuse is √2 times the length of a leg.
- If each leg of a 45-45-90 triangle has a length of 5√2, the length of the hypotenuse is 10.
- In a 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg.
- If the shorter leg of a 30-60-90 triangle has a length of 3, the length of the hypotenuse is 6.
Right Triangle Problems
- In a right triangle, if one leg has a length of 12 and the hypotenuse has a length of 13, the length of the other leg can be calculated using the Pythagorean theorem.
- The length of the other leg in this case is √(13^2 - 12^2) = √(169 - 144) = √25 = 5.
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