Final answer:
The Pythagorean Theorem relates the length of the legs of a right triangle (labeled a and b) with the hypotenuse (labeled c). To compare triangles and form one side of the theorem, set up and solve proportions. For example, if a = 3 and b = 4, the proportion would be a / 3 = b / 4. Cross multiply, simplify, and then use the equation a² + b² = c² to find the value of c.
Step-by-step explanation:
The Pythagorean Theorem relates the length of the legs of a right triangle, labeled a and b, with the hypotenuse, labeled c. The relationship is given by: a² + b² = c². To compare the triangles and form one side of the Pythagorean Theorem, we can use the method of setting up and solving proportions:
- Set up the proportions by comparing corresponding sides of the triangles.
- Cross multiply the ratios.
- Simplify the resulting equation.
For example, suppose we have two triangles with side lengths a = 3 and b = 4, and we want to find the hypotenuse c. We can set up the proportion as a / 3 = b / 4. Cross multiplying gives us 4a = 3b. Simplifying this equation, we get 4(3) = 3(4). Therefore, the simplified equation is 12 = 12, which is true. This confirms that the triangles are proportional.
Using this information, we can form one side of the Pythagorean Theorem, which is a² + b² = c². In this case, it would be 3² + 4² = c², which simplifies to 9 + 16 = c². Finally, we can use the equation to find the value of c by taking the square root of both sides: c = √(9 + 16).