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In the 30-60-90 triangle below, side s has a length of and side r has a length of .

Check the final answer first, then review the worked steps.

Problem

In the 30-60-90 triangle below, side s has a length of ____ and side r has a length of ____.

Answer

\(s = 6\sqrt{3}, r = 6\)

Step-by-step solution

  1. Identify the triangle type: The problem states that the triangle is a 30-60-90 triangle. This is a special type of right triangle with specific side length ratios.
  1. Recall the side length ratios of a 30-60-90 triangle: In a 30-60-90 triangle, if the side opposite the 30-degree angle (the shortest leg) is $x$, then the side opposite the 60-degree angle (the longer leg) is $x\sqrt{3}$, and the side opposite the 90-degree angle (the hypotenuse) is $2x$.
  1. Identify the given information: From the image, we can see that the hypotenuse (opposite the 90-degree angle) has a length of 12. The side opposite the 30-degree angle is labeled $r$, and the side opposite the 60-degree angle is labeled $s$.
  1. Set up an equation using the hypotenuse: According to the 30-60-90 triangle ratios, the hypotenuse is $2x$. We are given that the hypotenuse is 12. So, we can write the equation: $$2x = 12$$.
  1. Solve for x: Divide both sides of the equation by 2: $$x = \frac{12}{2} = 6$$. This value of $x$ represents the length of the side opposite the 30-degree angle, which is $r$.
  1. Determine the length of side r: Since $x$ is the length of the side opposite the 30-degree angle, $r = x$. Therefore, $r = 6$.
  1. Determine the length of side s: The side opposite the 60-degree angle is $s$, and its length is $x\sqrt{3}$. Substitute the value of $x$ we found: $$s = 6\sqrt{3}$$.
  1. State the lengths of sides s and r: Side $s$ has a length of $6\sqrt{3}$, and side $r$ has a length of 6.