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Geometry for Beginners – How to Identify and Use the 45 Right
Welcome to Geometry for Beginners. Success in geometry is highly 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 area of a sphere, we need to have the measurements needed for formulas. Having shortcuts to enable quick determination of these metrics can be a huge time saver. The 45 right “special triangle” gives us such a shortcut.
I can tell that you are just out of breath in anticipation of knowing this shortcut. Its good! The “need to understand” is part of what will lead you to success in math and everything else.
The 30-60 right triangle, covered in another article, and the 45 right triangle are “special” because no matter how big or small these triangles are, the three sides have a special relationship or relationship that is ALWAYS the same. We can use this ever-existing relationship to find the missing secondary measures without having to use the Pythagorean theorem; or we can determine whether a given triangle is or is not one of these triangles.
Another “always” in geometry is that pictures help understanding. Unfortunately, I don’t have the ability to include diagrams in articles. Therefore, you need to draw the necessary diagram while I describe what to draw. Start by drawing the capital letter L, but make sure the two “legs” are perpendicularr (forming a right or 90 degree angle) and are also the same length. Now draw the line segment that connects the ends of the two legs. You should now see a right-angled triangle with the bottom left right angle and both legs of equal length.
A right triangle 45 is also called a 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 all three angles of a triangle add up to 180 degrees, having one right angle tells us that the other two angles add up to 90 degrees. Since they are equal, they should each have a measurement of 45 degrees. On your drawing, place these angle measurements 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 a isosceles right triangle. So if you have a right triangle with both legs equal or both non-right angles equal, then the triangle MUST be a 45° right triangle.
Now we need to find out the relationship of the sides. To do this, we will 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 lower leg as having a measurement of 5 units. (There’s nothing special about 5 other than the fact that I like it.) Are you now able to label any other side? Certainly! The other leg should also have a measurement of 5. Label this side as well. we now have a 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 to find the missing side of a right triangle of which we already know two 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 (rewritten 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. So, from shortest to longest, the three sides have a ratio of 5:5:5 sqrt 2. This last term is reads as “5 times the square root of 2.”
It is no coincidence 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 sides in a right triangle of 45 is always an :a:a square 2.
We can use this report 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 measurement of the hypotenuse does not end in sqrt 2. For this situation, we need to do some algebra. For example: if the hypotenuse is 3, we need to write a small equation using the form a sqrt 2 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 the numerator and denominator by sqrt 2 eliminates the radical in the denominator, which some teachers require. So 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 6, 7, 6 sqrt 2 is NOT a 45 right triangle because the two shorter sides are not are not equal.
In conclusion, just like with 30-60 right triangles, knowing the relationship for a 45 right triangle and knowing when you can use it removes the need to use the Pythagorean theorem. A shortcut is good!
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