Special Right Triangles: 45-45-90

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Questions and Answers

In a 45-45-90 triangle, if one leg has a length of $7$, what is the length of the hypotenuse?

  • 14
  • $7\sqrt{2}$ (correct)
  • 7
  • 7$\sqrt{3}$

What is the length of each leg of a 45-45-90 triangle if the hypotenuse has a length of $10\sqrt{2}$?

  • 20
  • 10 (correct)
  • 5
  • 5$\sqrt{2}$

In a 30-60-90 triangle, the shortest side has a length of $3$. What is the length of the hypotenuse?

  • 3
  • 6$\sqrt{3}$
  • 6 (correct)
  • 3$\sqrt{3}$

What is the length of the longer leg in a 30-60-90 triangle if the shortest side has a length of $5$?

<p>5$\sqrt{3}$ (D)</p> Signup and view all the answers

In a 30-60-90 triangle, the hypotenuse has a length of $12$. What is the length of the short leg?

<p>6 (B)</p> Signup and view all the answers

If the longer leg of a 30-60-90 triangle has a length of $9\sqrt{3}$, what is the length of the shorter leg?

<p>9 (D)</p> Signup and view all the answers

The longer leg of a 30-60-90 triangle has a length of $6$. What is the length of the hypotenuse?

<p>4$\sqrt{3}$ (D)</p> Signup and view all the answers

A right triangle has two equal angles that are each $45$ degrees. If one of the equal sides has a length of $x+2$, what is the length of the hypotenuse?

<p>$(x+2)\sqrt{2}$ (A)</p> Signup and view all the answers

A triangle has angles measuring 30, 60, and 90 degrees. If the side opposite the 60-degree angle is $5\sqrt{3}$, find the length of the side opposite the 30-degree angle.

<p>5 (B)</p> Signup and view all the answers

If the hypotenuse of a right triangle is twice the length of one of its legs, what are the measures of the non-right angles in the triangle?

<p>30 and 60 degrees (A)</p> Signup and view all the answers

Flashcards

Special Right Triangles

Right triangles with specific angle measures that simplify calculations. Common types are 45-45-90 and 30-60-90 triangles.

45-45-90 Triangle

An isosceles right triangle with angles of 45, 45, and 90 degrees. Side ratio: leg : leg : hypotenuse = x : x : x√2.

Hypotenuse (45-45-90)

In a 45-45-90 triangle, multiply the leg length by √2.

Leg Length (45-45-90)

In a 45-45-90 triangle, divide the hypotenuse length by √2 (or multiply by √2/2).

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30-60-90 Triangle

A right triangle with angles of 30, 60, and 90 degrees. Side ratio: short leg : long leg : hypotenuse = x : x√3 : 2x.

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Hypotenuse (30-60-90)

In a 30-60-90 triangle, multiply the short leg length by 2.

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Long Leg (30-60-90)

In a 30-60-90 triangle, multiply the short leg length by √3.

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Short Leg (30-60-90)

In a 30-60-90 triangle, divide the hypotenuse length by 2.

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Short Leg from Long Leg (30-60-90)

In a 30-60-90 triangle, divide the long leg length by √3 (or multiply by √3/3).

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Pythagorean Theorem

The theorem stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides: a² + b² = c².

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Study Notes

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