THE PYTHAGOREAN THEOREM The Pythagorean Theorem, also known as Pythagoras's theorem, is a relation in Euclidean geometry among the three sides of a right triangle.
A right triangle has one angle that is 90째 (i.e., a right angle). The side opposite the right angle is called the hypotenuse, often labeled c, while the other two sides of the triangle (legs) are often labeled a and b.
For a right triangle with legs a and b and hypotenuse c: a2 + b2 = c2 Note: Which leg is a or b is irrelevant in the use of the Pythagorean Theorem. Note: The Pythagorean Theorem only works for right triangles. Putting the formula into words, the square of the length of one leg plus the square of the length of the other leg equals the square of the long-side opposite the right angle.
THE PYTHAGOREAN THEOREM EXERCISES Given any two sides of a right triangle, it is possible to calculate what the length of the third side must be. For example: What is the length of side c?
Pythagorean Theorem: a2 + b2 = c2 a = 3, b = 4, c = ? 32 + 42 = c2 9 + 16 = c2 25 = c2 c=5 Note: When taking the square root, normally one must take the positive and negative value (i.e., Âą5). However, since a negative length has no meaning when solving for the length of a side, the negative value is ignored.
Step By Step Examples of Using the Pythagorean Theorem Example 1 (solving for the hypotenuse) Use the Pythagorean Theorem to determine the length of X Step 1 Identify the legs and the hypotenuse of the right triangle. The legs have length '6 and '8'. 'X' is the hypotenuse because it is opposite the right angle. See Picture Step 2 Substitute values into the formula (remember 'c' is the hypotenuse)
A2 + B2 = C2 62 + 82 = X2 Step 3 Solve for the unknown
Example 2 (solving for a Leg)
'c' is the hypotenuse) A2 + B2 = C2 x2 + 242 = 262 Step 3 Solve for the unknown
1. Find the length of X
Use the Pythagorean the length of X Step 1 Identify the legs and right triangle. The legs have length hypotenuse because angle. See Picture Step 2 Substitute values into
Theorem to determine the hypotenuse of the '24’ and 'X'. '26' is the it is opposite the right the formula (remember
2. A 10 m long ladder is leaning against a wall. The bottom of the ladder is 6 m from the base of where the wall meets the ground. At what height from the ground does the top of the ladder lean against the wall? 3. Determine the side of an equilateral triangle whose perimeter is equal to a square of side 12 cm. Are their areas equal? 4. Determine the area of the square inscribed in a circle with a circumference of 18.84 cm. 5. The perimeter of an isosceles trapezoid is 110 m and the bases are 40 and 30 m in length. Calculate the length of the non-parallel sides of the trapezoid and its area. 6. A regular hexagon of side 4 cm has a circle inscribed and another circumscribed around its shape. Find the area enclosed between these two concentric circles. 7. A chord of 48 cm is 7 cm from the center of a circle. Calculate the area of the circle. 8. The legs of a right triangle inscribed in a circle measure 22.2 cm and 29.6 cm. Calculate the circumference and the area of the circle.