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What is the hypotenuse length of a 45°-45°-90° triangle with legs measuring 8 units each?
What is the hypotenuse length of a 45°-45°-90° triangle with legs measuring 8 units each?
For a 30°-60°-90° triangle with a hypotenuse of length 32, what is the length of the shorter leg?
For a 30°-60°-90° triangle with a hypotenuse of length 32, what is the length of the shorter leg?
If a 30°-60°-90° triangle has a longer leg of 20, what is the length of the shorter leg?
If a 30°-60°-90° triangle has a longer leg of 20, what is the length of the shorter leg?
In a 45°-45°-90° triangle with a hypotenuse of length 28, what are the lengths of the legs?
In a 45°-45°-90° triangle with a hypotenuse of length 28, what are the lengths of the legs?
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What is the value of the longer leg in a 30°-60°-90° triangle where the shorter leg measures 9√3?
What is the value of the longer leg in a 30°-60°-90° triangle where the shorter leg measures 9√3?
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For a 30°-60°-90° triangle, if the shorter leg measures x, what is the formula for the hypotenuse?
For a 30°-60°-90° triangle, if the shorter leg measures x, what is the formula for the hypotenuse?
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What is the hypotenuse length of a 45°-45°-90° triangle if one leg is measured to be x?
What is the hypotenuse length of a 45°-45°-90° triangle if one leg is measured to be x?
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If a 30°-60°-90° triangle has a hypotenuse of length 30, what is the length of the longer leg?
If a 30°-60°-90° triangle has a hypotenuse of length 30, what is the length of the longer leg?
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What is the length of the altitude drawn to the hypotenuse in a right triangle with legs of lengths 6 and 8 and a hypotenuse of length 10?
What is the length of the altitude drawn to the hypotenuse in a right triangle with legs of lengths 6 and 8 and a hypotenuse of length 10?
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In right triangle XYZ, which triangles are similar due to the altitude WX drawn from the right angle?
In right triangle XYZ, which triangles are similar due to the altitude WX drawn from the right angle?
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Calculate the altitude length x in a right triangle with a hypotenuse of length 39 and segments of the hypotenuse measuring 3 and 36.
Calculate the altitude length x in a right triangle with a hypotenuse of length 39 and segments of the hypotenuse measuring 3 and 36.
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What is the geometric mean of 16 and 27?
What is the geometric mean of 16 and 27?
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If one leg of a right triangle is 18 and the longer segment of the hypotenuse is 28, what would the altitude x be?
If one leg of a right triangle is 18 and the longer segment of the hypotenuse is 28, what would the altitude x be?
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What is the altitude in a right triangle with segments of the hypotenuse measuring 55 and 8, and a hypotenuse of 63?
What is the altitude in a right triangle with segments of the hypotenuse measuring 55 and 8, and a hypotenuse of 63?
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A right triangle has legs of 20 and 48. What is its hypotenuse length?
A right triangle has legs of 20 and 48. What is its hypotenuse length?
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In a right triangle with legs of lengths 13.2 and 26, what is the length of the hypotenuse?
In a right triangle with legs of lengths 13.2 and 26, what is the length of the hypotenuse?
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What is the geometric mean of the numbers 6 and 30?
What is the geometric mean of the numbers 6 and 30?
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If the altitude is the geometric mean of the two segments with lengths 8 and 12, what is the length of the altitude?
If the altitude is the geometric mean of the two segments with lengths 8 and 12, what is the length of the altitude?
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In a right triangle, if the hypotenuse has a length of 25 and one leg corresponds to a segment length of 7, what is the length of that leg?
In a right triangle, if the hypotenuse has a length of 25 and one leg corresponds to a segment length of 7, what is the length of that leg?
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What is the value of the altitude in a right triangle if it splits the hypotenuse into segments of lengths 5 and 37?
What is the value of the altitude in a right triangle if it splits the hypotenuse into segments of lengths 5 and 37?
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If a right triangle has segments of the hypotenuse that are 15 and 18, what is the length of the altitude?
If a right triangle has segments of the hypotenuse that are 15 and 18, what is the length of the altitude?
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For a right triangle split into segments of lengths 2 and 24, what is the length of one leg?
For a right triangle split into segments of lengths 2 and 24, what is the length of one leg?
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In a right triangle where the hypotenuse measures 21, the segments are 11.2 and what length to find the opposite leg?
In a right triangle where the hypotenuse measures 21, the segments are 11.2 and what length to find the opposite leg?
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What is the Pythagorean theorem used for?
What is the Pythagorean theorem used for?
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In a 30°-60°-90° triangle with a hypotenuse of length 11, what is the length of the longer leg?
In a 30°-60°-90° triangle with a hypotenuse of length 11, what is the length of the longer leg?
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If a triangle has sides of length 8, 15, and 17, how can it be classified?
If a triangle has sides of length 8, 15, and 17, how can it be classified?
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What would be the length of the hypotenuse if the legs of a right triangle are 12 and 16?
What would be the length of the hypotenuse if the legs of a right triangle are 12 and 16?
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What is the hypotenuse length in a 45°-45°-90° triangle if the length of each leg is $9√2$?
What is the hypotenuse length in a 45°-45°-90° triangle if the length of each leg is $9√2$?
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If two sides of a triangle measure 10 and 24, which of the following could represent the third side?
If two sides of a triangle measure 10 and 24, which of the following could represent the third side?
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What is the relationship between the lengths of the legs in a 45°-45°-90° triangle?
What is the relationship between the lengths of the legs in a 45°-45°-90° triangle?
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What is the shorter leg's length in a 30°-60°-90° triangle where the longer leg is $6$?
What is the shorter leg's length in a 30°-60°-90° triangle where the longer leg is $6$?
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In a 30°-60°-90° triangle, if the shorter leg is $x$ and the hypotenuse is $32$, what is the longer leg's length?
In a 30°-60°-90° triangle, if the shorter leg is $x$ and the hypotenuse is $32$, what is the longer leg's length?
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In a 30°-60°-90° triangle, if the shorter leg measures 4√15, what is the length of the hypotenuse?
In a 30°-60°-90° triangle, if the shorter leg measures 4√15, what is the length of the hypotenuse?
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What is the classification of a triangle with sides measuring 3, 4, and 5?
What is the classification of a triangle with sides measuring 3, 4, and 5?
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What height does a 16-foot ladder reach against a wall if the base is 5 feet from the wall?
What height does a 16-foot ladder reach against a wall if the base is 5 feet from the wall?
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What is the length of the longer leg in a 30°-60°-90° triangle if the shorter leg is 6?
What is the length of the longer leg in a 30°-60°-90° triangle if the shorter leg is 6?
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What is the relationship of the legs in a 45°-45°-90° triangle with legs of length $x$?
What is the relationship of the legs in a 45°-45°-90° triangle with legs of length $x$?
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If a triangle has side lengths of 7, 24, and 25, what type of triangle is it?
If a triangle has side lengths of 7, 24, and 25, what type of triangle is it?
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If the hypotenuse of a 45°-45°-90° triangle is 39√2, what is the length of each leg?
If the hypotenuse of a 45°-45°-90° triangle is 39√2, what is the length of each leg?
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For a 30°-60°-90° triangle with a shorter leg of $12$, what is the hypotenuse's length?
For a 30°-60°-90° triangle with a shorter leg of $12$, what is the hypotenuse's length?
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In a right triangle where the hypotenuse is 29 and one leg is 12, what is the length of the other leg?
In a right triangle where the hypotenuse is 29 and one leg is 12, what is the length of the other leg?
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If a right triangle's altitude divides the hypotenuse into segments $m$ and $n$, how is the altitude $h$ related to these segment lengths?
If a right triangle's altitude divides the hypotenuse into segments $m$ and $n$, how is the altitude $h$ related to these segment lengths?
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What is the perimeter of a square with a side length of 28?
What is the perimeter of a square with a side length of 28?
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In a 30°-60°-90° triangle where the longer leg is $14√3$, what is the length of the shorter leg?
In a 30°-60°-90° triangle where the longer leg is $14√3$, what is the length of the shorter leg?
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If a saw ramp makes a 30° angle with the ground and reaches a height of 37.5 inches, what is the length of the ramp in feet?
If a saw ramp makes a 30° angle with the ground and reaches a height of 37.5 inches, what is the length of the ramp in feet?
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How do you find the length of the longer leg in a 30°-60°-90° triangle if you only know the hypotenuse?
How do you find the length of the longer leg in a 30°-60°-90° triangle if you only know the hypotenuse?
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What would the perimeter of a triangle be if it is a 30°-60°-90° triangle with a shorter leg of 4√15 and a hypotenuse of 8√15?
What would the perimeter of a triangle be if it is a 30°-60°-90° triangle with a shorter leg of 4√15 and a hypotenuse of 8√15?
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Study Notes
Right Triangle Trigonometry
- Trigonometric ratios: Sine, cosine, and tangent are ratios of the sides of a right triangle relative to a given angle.
- Sine (sin): The ratio of the side opposite the angle to the hypotenuse.
- Cosine (cos): The ratio of the side adjacent to the angle to the hypotenuse.
- Tangent (tan): The ratio of the side opposite the angle to the side adjacent to the angle.
Pythagorean Theorem
- Formula: a² + b² = c²
-
Variables:
- a and b are the legs of the right triangle.
- c is the hypotenuse (the side opposite the right angle).
- Application: Used to find the missing side of a right triangle when two sides are known.
Special Right Triangles
-
45°-45°-90° Triangle:
- Legs are congruent.
- Hypotenuse is √2 times the length of each leg.
-
30°-60°-90° Triangle:
- Shorter leg is opposite the 30° angle.
- Longer leg is opposite the 60° angle.
- Hypotenuse is twice the length of the shorter leg.
Right Triangle Similarity
- Altitude Theorem: If an altitude is drawn to the hypotenuse of a right triangle, the two triangles formed are similar to the original triangle and to each other.
- Leg Theorem: In a right triangle, the altitude drawn to the hypotenuse is the geometric mean between the segments of the hypotenuse. Each leg of the triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Geometric Mean
- Definition: The geometric mean of two positive numbers a and b is the positive number x such that x² = ab, or x = √(ab).
Angle of Elevation
- Definition: The angle of elevation is the angle formed between a horizontal line of sight and the line of sight to an object above that horizontal line.
Angle of Depression
- Definition: The angle of depression is the angle formed between a horizontal line of sight and the line of sight to an object below that horizontal line. The angle of depression is congruent to the angle of elevation.
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Description
Explore the fundamental concepts of right triangle trigonometry, including trigonometric ratios like sine, cosine, and tangent. Additionally, delve into the Pythagorean theorem and special right triangles, learning their properties and applications. Test your knowledge with this comprehensive quiz on these key geometric principles.