Question

use the pythagorean theorem to derive the equation of the circle centered at the origin with a diameter of 14. to which triangle should you apply the pythagorean theorem? what is the equation of the circle? (x^{2}+y^{2}=49) ((7x)^{2}+(7y)^{2}=49) ((x + 7)^{2}+(y + 7)^{2}=49) ((x - 7)^{2}+(y - 7)^{2}=49)

Explanation:

Step1: Recall circle - point relationship

A circle centered at the origin \((0,0)\) with radius \(r\) has a point \((x,y)\) on its circumference. The distance from the origin \((0,0)\) to the point \((x,y)\) is the radius. We can form a right - triangle with legs of lengths \(|x|\) and \(|y|\) and hypotenuse equal to the radius \(r\).

Step2: Apply Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). In our case, \(a = x\), \(b = y\), and \(c=r\). So, \(x^{2}+y^{2}=r^{2}\).

Step3: Find the radius

Given the diameter \(d = 14\), the radius \(r=\frac{d}{2}=\frac{14}{2}=7\).

Step4: Substitute the radius into the equation

Substitute \(r = 7\) into \(x^{2}+y^{2}=r^{2}\), we get \(x^{2}+y^{2}=49\).
The triangle to which we should apply the Pythagorean theorem is the right - triangle with vertices \((0,0)\), \((x,0)\), and \((x,y)\) (or \((0,0)\), \((0,y)\), and \((x,y)\)).

Answer:

The triangle to apply the Pythagorean theorem: The right - triangle with vertices \((0,0)\), \((x,0)\), and \((x,y)\) (or \((0,0)\), \((0,y)\), and \((x,y)\)); The equation of the circle: \(x^{2}+y^{2}=49\)