Question

the pythagorean theorem can be used to: a. solve quadratic equations b. determine the distance between two points on a coordinate plane c. calculate the circumference of a circle d. find the surface area of a sphere

Explanation:

Step1: Recall the Pythagorean Theorem

The Pythagorean Theorem is \(a^{2}+b^{2}=c^{2}\) for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\).

Step2: Analyze the distance formula

The distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) on a coordinate plane is derived from the Pythagorean Theorem. If we consider the horizontal distance \(\Delta x=x_2 - x_1\) and vertical distance \(\Delta y=y_2 - y_1\) as the legs of a right - triangle and the distance \(d\) as the hypotenuse.

Step3: Analyze other options

• Option a: Quadratic equations are solved using methods like factoring, completing the square, or the quadratic formula (\(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)), not the Pythagorean Theorem.
• Option c: The circumference of a circle is \(C = 2\pi r\) (where \(r\) is the radius), which has no relation to the Pythagorean Theorem.
• Option d: The surface area of a sphere is \(S = 4\pi r^{2}\), which is not related to the Pythagorean Theorem.

Answer:

b. Determine the distance between two points on a coordinate plane