Question

kyle states he can find the distance between two points on the coordinate plane by creating a triangle and using the pythagorean theorem. ava states she can find the distance between the two points using only the coordinates of each point and the distance formula. which student is correct, and why?
options:

• only ava is correct. her method of using the coordinates of each point and the distance formula would work. kyle may not be able to create a right triangle from the set of two points, and therefore might not be able to solve with the pythagorean theorem.
• both students are correct. the pythagorean theorem is derived from the distance formula, so both methods are correct.
• only kyle is correct. his method of creating a triangle and using the pythagorean theorem would work. ava is correct that she could use the distance formula, but she would need more than just the coordinates of each point to do so.
• both students are correct. the distance formula is derived from the pythagorean theorem, so both methods are correct.

Explanation:

The Distance Formula \( d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \) is derived from the Pythagorean Theorem (\( a^2 + b^2 = c^2 \)). To find the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\), we can form a right triangle where the horizontal leg length is \( |x_2 - x_1| \), the vertical leg length is \( |y_2 - y_1| \), and the distance between the points is the hypotenuse \( c \). Using the Pythagorean Theorem, \( c=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), which is the Distance Formula. So both methods (using the Pythagorean Theorem by creating a triangle and using the Distance Formula with coordinates) are valid because the Distance Formula comes from the Pythagorean Theorem. Thus, both students are correct.

Answer:

Both students are correct. The Distance Formula is derived from the Pythagorean Theorem, so both methods are correct.