Question

select the correct answer from each drop - down menu. valeries work deriving the distance formula for length d is shown. evaluate her work and explain any errors. she chose a point that a right angle. she wrote the length of each of the legs. she in the equation.

Explanation:

To derive the distance formula, we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). In a coordinate - plane, if we have two points \((x_1,y_1)\) and \((x_2,y_2)\), the horizontal distance (one leg of the right - triangle) is \(|x_2 - x_1|\) and the vertical distance (the other leg of the right - triangle) is \(|y_2 - y_1|\). The distance \(d\) between the two points is the hypotenuse of the right - triangle.

The Pythagorean theorem gives \(d^{2}=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\), and then \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)

1. For the first drop - down:

• The construction of the right - triangle for the distance formula requires that the point forms a right - angle. Since we can form a right - triangle with the two points \((x_1,y_1)\) and \((x_2,y_2)\) by creating a horizontal and a vertical line (which are perpendicular), she chose a point that does not form a right - angle. The correct right - triangle legs should be based on the horizontal and vertical differences. If we consider the standard distance formula derivation, the horizontal leg length is \(|x_2 - x_1|\) (a non - negative quantity representing the horizontal change) and the vertical leg length is \(|y_2 - y_1|\) (a non - negative quantity representing the vertical change). In her work, the way she labeled \((x_2 - x_1)\) and \((y_2 - y_1)\) as legs is incorrect in terms of forming a right - triangle (the right - angle should be formed by a horizontal and a vertical line, not in the way she has it in the graph).

1. For the second drop - down:

• The length of a line segment in the coordinate plane for the horizontal leg (parallel to the \(x\) - axis) is \(|x_2 - x_1|\) and for the vertical leg (parallel to the \(y\) - axis) is \(|y_2 - y_1|\). She incorrectly wrote the length of each of the legs. The values \((x_2 - x_1)\) and \((y_2 - y_1)\) can be negative (depending on the order of the points), but the length of a side of a triangle (a geometric quantity) must be non - negative.

1. For the third drop - down:

• Using the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse (distance \(d\) between two points) and \(a\) and \(b\) are the legs. She did not take the square root in the equation. The correct formula is \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), but she wrote \(d=(x_2 - x_1)^{2}+(y_2 - y_1)^{2}\)

Answer:

1. does not form
2. incorrectly
3. did not take the square root