45-45-90 Triangle: Theorem, Rules & Formula

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  • 0:05 Rules of a 45-45-90 Triangle
  • 1:48 Formula and Theorem
  • 4:01 Lesson Summary
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Lesson Transcript
Instructor: Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

This lesson will teach you about one of the special right triangles, the 45-45-90 triangle. You will learn the formulas for calculating the lengths of the sides of this type of triangle. After the lesson, you will be able to test your knowledge with a quiz.

Rules of a 45-45-90 Triangle

When we are talking about a 45-45-90 triangle, those numbers represent the measures of the angles of that triangle. So, it means the triangle has two 45-degree angles and one 90-degree angle.

Let's do a little origami to make one of these triangles! Don't worry; you don't have to make a swan! Origami is usually made using a square piece of paper. Imagine you take a square and fold it so that two of the corners end up on top of each other, like this:

Folding a square diagonally to make a triangle

We placed corner B on corner D and made a crease diagonally through corners A and C. Now we have made a triangle. Since we did not cut anything off the paper, we know several things about that triangle. The angle at D is a right angle (measures 90 degrees) because the square had four right angles. The angles A and C must each be 45 degrees because the right angles there were folded exactly in half. Since we folded exactly through the corners, the lengths of the sides AD and DC did not change from what they were in the square. Therefore, we know that the sides AD and DC must be of equal length because on the square, all of the sides were of equal length.

Doing this little folding exercise, we have discovered that every 45-45-90 triangle has two sides with the same length. Those sides are called the legs of the triangle. The hypotenuse (the side opposite of the right angle) will always have a length longer than the legs. In the next two sections, we will talk about the formula to calculate the length of the hypotenuse and the theorem for 45-45-90 triangles.

Formula and Theorem

The relationship between the three sides of any kind of right triangle is given by the Pythagorean Theorem. The formula for the Pythagorean Theorem is a2 + b2 = c2. The rule for using this formula is that c must stand for the hypotenuse. It does not matter which of the two sides you call a and which you call b.

Now, let's take a look at how the Pythagorean Theorem works for a generic 45-45-90 triangle. That way, we can figure out a formula that relates the length of the hypotenuse to the length of the leg of any 45-45-90 triangle.

In triangle FTY, we labeled the hypotenuse c according to the rule of the Pythagorean Theorem. The other two sides, as we mentioned above, have the same length, so we do not have to label one of them a and the other b. Instead, we can assign both legs the same variable. Let's use g.

45-45-90 triangle with hypotenuse c and legs g

Now, let's fill in the Pythagorean Theorem formula and then do a little algebra to clean things up.

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