Isosceles Right Triangles

The isosceles 45-45-90 special right triangle’s properties are used to find two missing sides when one side and the angles, 45°, 45°, and 90° are given.  The corresponding sides have a general measure of x, x, and x √2.

Example:  Jan is marking off her summer garden. Each corner of the rectangular garden is marked off with a 3 foot string, creating an isosceles right triangle where a tomato plant will be placed. What are the dimensions of the isosceles triangle where the tomato plant will be placed? If the four tomato plants will be fenced in, how much fencing is needed?

• The first step is to sketch a Interactive button. Assistance may be required. ___?___ picture or diagram
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• Since the garden is a rectangle, the triangles are Interactive button. Assistance may be required. ___?___ right triangles.
• Since the right triangle is an isosceles right triangle, the angle measures are Interactive button. Assistance may be required. ___?___ 45°, 45°, and 90° .
  
• The length of the string is Interactive button. Assistance may be required. ___?___ 3 feet.
  
• The string is the Interactive button. Assistance may be required. ___?___ hypotenuse of the 45-45-90 special right triangle.
• In general, the dimensions of the sides of a 45-45-90 special right triangle are Interactive button. Assistance may be required. ___?___ x, x, and x √2.
• Label the triangle.
  
• The general dimension for the hypotenuse is Interactive button. Assistance may be required. ___?___ x √2 .
• Therefore, 3 = Interactive button. Assistance may be required. ___?___ x √2 .
• Solve for x to find the lengths of the legs.
  
  Interactive popup. Assistance may be required.

  Check Your Answer

  3 = x √2 ⇒ 3 over square root of 2 3 √2 = x times the square root of 2 over the sqaure root of 2 x√2 √2 ⇒ x = 3 over square root of 2 3 √2 × the square root of 2 over the square root of 2 √2 √2 = 3 times the square root of 2 over 2 3 √2 2
• Both legs of the special triangle have the same length, Interactive button. Assistance may be required. ___?___ 3 times the square root of 2 over 2 3 √2 2 or 3 times the square root of 2 over 2 3 √2 2 = 2.12 feet.
• There are Interactive button. Assistance may be required. ___?___ four triangles in the garden for tomato plants.
• The amount of fencing needed is Interactive button. Assistance may be required. ___?___ 4(2.12 + 2.12 + 3) = 4(7.24) = 28.96
  or approximately 29 feet.