Right Triangles

Right Triangles

The right triangle has been used in trades for thousands of years. Ancient Egyptians found that they could always get a square corner using the 3-4-5 right triangle. (Carpenters still use the 3-4-5 triangle to square corners.) Later, a Greek mathematician named Pythagoras developed a formula to find the side lengths for any right triangles.

Pythagoras treated each side of
a right triangle as if it was a
side of a square. He found that
the total area of the two smaller
squares was equal to the area of
the largest square for every right
triangle.

The area of any square is found
by squaring one of the sides.
The areas of these squares are
are 3^2 , 4^2 , and 5^2 . If the
two shorter sides are squared
and added together, the answer
equals the longer side squared:

The three sides of a right triangle are represented by the variables a, b, and c. The variable c always represents the longest side. You can write the formula used above as .

This is the formula of the Pythagorean Theorem.

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The longest side of a right triangle is always directly across from the 90 degree angle. This side is called the hypotenuse.

The other two sides are often called
the legs. For now, it really doesn't
matter which one of the legs you call
a or b, as long as you make one a
and the other b.

Because the 3-4-5 right triangle is simple to work with,
let's use it to show how the Pythagorean formula
verifies that a triangle is a right triangle.

Since the numbers on both sides of the = mark are the same, the 3-4-5 triangle is a right triangle.

Please note: The Pythagorean formula is a rule! To verify that a triangle is a right triangle, use the formula.
If the numbers on each side of the = sign are equal, then the triangle is a right triangle.

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Right triangle practice: Check these triangles to see which ones are right triangles.

           a     b     c               a       b       c

     (1)   6     8     10        (6)  9.99   13.32   16.65
     (2) .48   .64     .8        (7)  4.5       6     7.5
     (3)   1     1      1        (8)   7       10     13
     (4)   8     6     10        (9)   9      2.5     13
     (5)  120   160   200       (10)   2     2.6666  3.3333

Answers.

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Note:

Lower case letters (a, b, c)
are used as variables for the sides.

Upper case letters (A, B, C)
are used for the angles opposite
those sides.