The sides of
the right triangle below have lengths of 3, 4, and 5 units.
The relationship among these lengths forms the basis for one of the most famous
theorems in mathematics.
The two
sides that form the right angle are called the legs. In the triangle above, the lengths of the
legs are 3 units and 4 units. The side opposite the right angle is called the hypotenuse. The
hypotenuse of this triangle has a length of 5 units.
The squares
drawn along each side of the triangle illustrate the Pythagorean Theorem
geometrically. Study the areas of the squares. Do you notice a relationship
between them? The area of the larger square is equal to the total area of the
two smaller squares.
25 = 9 + 16
52 = 32 + 42
This
relationship is true for any right triangle and is called the Pythagorean Theorem.
1.
Find the length of the hypotenuse of the right triangle.
Alternative Solutions:
The length of the hypotenuse is 17
feet.
You can also
use the Pythagorean Theorem to find the length of a leg of a right triangle.
Example
2.
Find the length of one leg of a right triangle if the length of the
hypotenuse is 14 meters and the length of the other leg is 6 meters. Round to
the nearest.
Alternative Solutions:
To the nearest tenth, the length of
the leg is 12.6 meters.
A converse of a theorem is
the reverse, or opposite, of the theorem. You can use the converse of the
Pythagorean Theorem to test whether a triangle is a right triangle.
Example
3. The measures of the three sides of a triangle are 5, 7, and 9. Determine whether this triangle is a right triangle.
Alternative Solutions:
Since c2 ≠ a2
+ b2, the triangle is not a right triangle.
Example
Carpentry Link
4. A carpenter checks whether the corners of a deck are square by
using a 3-4-5 system. He or she measures along one side in 3-foot
units and along the adjacent side in the same number of 4-foot
units. If the measure of the hypotenuse is the same number of
5-foot units, the corner of the deck is square. If
a corner is square,
it is formed by a right angle. So, the triangle is a right triangle.
Suppose a carpenter
measures along one side of a deck, a distance of 9 feet, and along the adjacent
side, a distance of 12 feet. The measure of the hypotenuse is 15 feet. Is the
corner of the deck square?
Alternative Solutions:
Sumber
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