Question

Unit 8: Right Triangles & Trigonometry Homework 2: Special Right Triangles...Please Help!

Answer 1

Answer:

• x = 8√3
• y = z = 12√2

Step-by-step explanation:

We presume you want the values of x, y, and z.

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There are two "special triangles" in geometry and trigonometry. They are the 30°-60°-90° right triangle that is half of an equilateral triangle, and the 45°-45°-90° isosceles right triangle that is half a square (cut by the diagonal).

The side ratios of these special triangles are relatively easy to remember. It is useful to memorize them.

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For the isosceles right triangle, the side lengths are the same. The Pythagorean theorem tells you that if they are both 1, then the hypotenuse is ...

  √(1²+1²) = √2

That is, the side lengths of the 45-45-90 triangle are in the ratio ...

  1 : 1 : √2

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For the triangle that is half an equilateral triangle, you know the hypotenuse is twice the length of the shortest side (since we got that short side by cutting a long side in half). Then the longer side can be found from the Pythagorean theorem:

  √(2²-1²) = √3

That is, the side lengths of the 30-60-90 triangle are in the ratio ...

  1 : √3 : 2

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In this problem, we're given the hypotenuse of a 30-60-90 triangle, so we know the short side of it (x) will be half that length:

  x = (16√3)/2

  x = 8√3

The hypotenuse of the 45-45-90 triangle will be √3 times x, so will be ...

  long side of small triangle = (√3)(8√3) = 24

The shorter sides of that 45-45-90 triangle will be this value divided by the square root of 2, so are ...

  y = z = 24/√2

We can multiply this by (√2)/(√2) to "rationalize the denominator".

  y = z = 12√2