Scalene Right Triangles

The scalene 30-60-90 special right triangle’s properties are used to find two missing sides when one side and the angles 30°, 60°, and 90° are given.

The corresponding sides have a general measure of x, 2x, and x √3.

Example:  A landscaper was planting trees. He had to secure braces to the trees to ensure they would grow straight. A brace is wrapped 4 feet above the ground around the tree. A rope is attached to the brace and to a stake put into the ground. The angle created between the ground and the rope attached to the stake has a measure of 60°. How far from the tree is the stake placed and how long is the rope?

• The first step is to sketch a Interactive button. Assistance may be required. ___?___ picture or diagram
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Known facts:

• The triangle is a Interactive button. Assistance may be required. ___?___ right triangle.
• The angles of the triangle are Interactive button. Assistance may be required. ___?___ 30, 60, and 90 degrees.
• The length of the side opposite the 60° angle is Interactive button. Assistance may be required. ___?___ 4 feet .
• This is the Interactive button. Assistance may be required. ___?___ longer leg of the 30-60-90 special right triangle.
• The shortest leg of a 30-60-90 right triangle is Interactive button. Assistance may be required. ___?___ half the length of the hypotenuse.
• In general, the legs and sides of a 30-60-90 right triangle are Interactive button. Assistance may be required. ___?___ x, 2x, and x √3 .
• Label this triangle:
  
  
  
• The side opposite the 60° angle is the Interactive button. Assistance may be required. ___?___ longest leg and has a general measure of x √3
  
• Therefore, 4 = Interactive button. Assistance may be required. ___?___ x √3 .
• The shortest leg is Interactive button. Assistance may be required. ___?___ x .
• Solve for x to find the distance the stake is from the tree. The process is Interactive button. Assistance may be required. ___?___ 4 = x√3 ⇒ x times the square root of 3 over the sqaure root of 3 x √3 √3 = 4 over the sqaure root of 3 4 √3 ⇒ x = 4 over the sqaure root of 3 4 √3 × The square root of 3 over the sqaure root of 3 √3 √3 = 4 times the square root of 3 over the sqaure root of 3 4 √3 √3 ⇒ x = 4 times the square root of 3 over the sqaure root of 3 4 √3 √3
• The stake is Interactive button. Assistance may be required. ___?___ 4 times the square root of 3 over 3 4 √3 3 feet from the tree.
• The hypotenuse of a 30-60-90 right triangle is Interactive button. Assistance may be required. ___?___ twice the length of the shortest side, 4 times the square root of 3 over 3 4 √3 3 feet.
• Find the hypotenuse in order to find the length of the rope. The answer is: Interactive button. Assistance may be required. ___?___ C = 4 times the square root of 3 over 3 4 √3 3 × 2 = 8 times the square root of 3 over 3 8 √3 3
• The rope is Interactive button. Assistance may be required. ___?___ 8 times the square root of 3 over 3 8 √3 3 feet long.
• Changing the radicals to decimals, the stake is Interactive button. Assistance may be required. ___?___ 2.31 feet from the tree and the brace is attached to the tree Interactive button. Assistance may be required. ___?___ 4.62 feet above the ground.