Welcome to Geometry for Beginners. Success in Geometry is heavily dependent on the ability to find missing measurements in order to evaluate formulas. Whether we need to determine if lines are parallel, find the height of a triangle, or find the surface area of a sphere, we must have the measurements needed for the formulas. Having shortcuts to allow quick determination of these measurements can be a huge time-saver. The 45 right "special triangle" gives us one such shortcut.
I can tell you are just breathless in anticipation of knowing this shortcut. That's good! The "need to understand" is part of what will lead you to success in math as well as in everything else.
Both the 30-60 right triangle, covered in another article, and the 45 right triangle are "special" because no matter how tiny or huge these triangles might be, the three sides have a special relationship or ratio that is ALWAYS the same. We can use this always existing relationship to find missing side measures without having to use the Pythagorean Theorem; or we can determine if a given triangle is or is not one of these triangles.
Another "always" in Geometry is that pictures help understanding. Unfortunately, I do not have the capability of including diagrams in articles. Consequently, you should draw the needed diagram as I describe what to draw. Start by drawing the capital letter L, but make sure the two "legs" are perpendicular (form a right or 90 degree angle) and are also the same length. Now, draw the line segment that connects the ends of both legs. You should now see a right triangle with the right angle on the lower left and both of its legs of equal length.
A 45 right triangle is also called an isosceles right triangle since it has two equal sides. An important property of isosceles triangles is that the angles opposite the equal sides are also equal. This means that for our diagram, the two non-right angles are equal in measure. Since the three angles of a triangle have a total of 180 degrees, then having one right angle tells us the other two angles total 90 degrees. Since they are equal, they must each have a measure of 45 degrees. On your drawing, place these angle measures inside the appropriate angles: 90, 45, and 45 degrees.
You have just created a 45 right triangle. Always remember that it is the same as an isosceles right triangle. Thus, if you have a right triangle with either both legs equal or both non-right angles equal, then the triangle MUST be a 45 right triangle.
Now, we need to discover the relationship of the sides. To do this, we are going to use specific values. (This is NOT a proof. It is a demonstration. If we followed the same procedure with variables, it would become a proof.)
Look at your drawing again. Let's label the base or bottom leg as having a measure of 5 units. (There is nothing special about 5 other than that I just like it.) Are you now able to label any other side? Certainly! The other leg must also have a measure of 5. Label that side as well. we now have part of our relationship. Since the legs are always the same, we could write their ratio as a:a. Put a's on your diagram under the 5's.
How do we find the missing side of a right triangle when we already know two of the sides? That's right! We use our old friend, the Pythagorean Theorem or c^2 = a^2 + b^2. For our example this becomes c^2 = 5^2 + 5^2 or c^2 = 25 + 25 or C^2 = 50. So c = sqrt 50. Since 50 can be factored (re-written as multiplication) using a perfect square, we can simplify this radical. Thus, sqrt 50 = sqrt (25 x 2) = sqrt 25 x sqrt 2 = 5 sqrt 2. Thus, from shortest to longest, the three sides have a ratio of 5:5:5 sqrt 2. This last term is read as "5 times the square root of 2."
It is not an accident that the hypotenuse (side opposite the right angle) is 5 sqrt 2. Doing a few more examples or following the same process with a variable will show you that the ratio of the sides in a 45 right triangle is always a:a:a sqrt 2.
We can use this ratio to find a missing side. For example, if we have a 45 right triangle with a leg of 13 units, then the other two sides must be 13 and 13 sqrt 2. This process is a little more difficult if the hypotenuse measure does not end with sqrt 2. For this situation, we have to do a little Algebra. For example: If the hypotenuse is 3, we need to write a small equation using the a sqrt 2 form for the hypotenuse: a sqrt 2 = 3. To solve for a, we need to divide both sides of this equation by sqrt 2. This gives us a = 3 / sqrt 2. Multiplying both numerator and denominator by sqrt 2 eliminates the radical in the denominator, which some teachers require. Thus, the final simplified answer is that a = 3 sqrt 2 / 2.
We can also use this ratio to determine if the triangle is a 45 right triangle. For example, a triangle with sides of 6, 7, 6 sqrt 2 is NOT a 45 right triangle because the two shorter sides are not equal.
In conclusion, just as with 30-60 right triangles, knowing the relationship for a 45 right triangle and recognizing when you can use it removes the need to use the Pythagorean Theorem. A shortcut is a good thing!
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