The right triangle comes along frequently in geometry. It is used over and over for examples, since it offers a readymade right angle, a hypotenuse, and other great parts. Finding the area of a right triangle is easy and fast.
After working your way through this lesson and video, you will be able to:
Let's create a right triangle, , with as the right angle. It measures and has the hypotenuse, or longest side, opposite it. That is side , long. It is a right triangle because it has a right angle, not because it is facing to the right.
Insert first drawing: right triangle, , is the right angle, ≈53.1, ∠S ≈36.8. Drawing oriented with sides and forming vertical and horizontal. Rotate for second drawing so is vertical and is horizontal. Rotate for third drawing so side is horizontal and introduce altitude from to Point on side . Triangle and altitude form figure CH.
The word "right" refers to the Latin word rectus, which means upright. A right angle shows one line or line segment upright from another; a right triangle has an upright angle.
The base of any triangle is the side used to create an altitude, or height, to an opposite vertex. The way our is sitting, you can easily see that side is the base for the altitude to . Side is the base for the altitude .
[insert second drawing]
For a right triangle, the sides adjacent to the right angle serve double duty as bases and altitudes, making the calculations of area really, really easy.
The altitude of any triangle is the perpendicular from a base (side) to the opposite vertex. For our we can rotate the figure one more time and place side on the horizontal, then construct an altitude from down to the base. The point where it intersects can be labelled . Now we have and Point ; !
The area of a right triangle is the measure of its interior space, in square units. For any triangle, the formula is:
For a right triangle, this is really, really easy to calculate using the two sides that are not the hypotenuse. In our we can use side , 24 yards, as the base, making side , 18 yards, the altitude:
If we rotate the right triangle and use side as the base, becomes the altitude, and you get the same answer!
Rotate it one more time and use side , 30 yards, as the base, with constructed altitude AH, which is 14.4 yards in height:
[insert third drawing]
All three sides, all three altitudes or heights, and all three answers the same! Remember the area is always in square units of the linear measurement (yards, in our triangle).
After reviewing this lesson, you are now able to identify a right triangle, manipulate a right triangle to find all its altitudes or heights, and recall and apply the formula ½ base x height to find the area of a right triangle.
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