Question

select the correct answer from each drop - down menu. explain how the distance of the line segment shown on the graph can be found. draw a right triangle where \\( \overline{st} \\) is the hypotenuse and the legs intersect at r. point r has coordinates of the length of \\( \overline{rs} \\) is, and the length of \\( \overline{rt} \\) is. using the pythagorean theorem, the length of \\( \overline{st} \\) is approximately

Explanation:

Step1: Find Coordinates of S and T

From the graph, point \( S \) is at \( (-2, 3) \) and point \( T \) is at \( (1, 5) \) (assuming grid squares are 1 unit each). To form a right triangle, we find point \( R \) by aligning horizontally and vertically from \( S \) and \( T \). So \( R \) should have the same \( x \)-coordinate as \( T \) and same \( y \)-coordinate as \( S \), so \( R = (1, 3) \).

Step2: Length of \( \overline{RS} \)

The length of \( \overline{RS} \) is the horizontal distance between \( S(-2, 3) \) and \( R(1, 3) \). Using the distance formula for horizontal line (\( y \)-coordinates same): \( |x_2 - x_1| = |1 - (-2)| = 3 \).

Step3: Length of \( \overline{RT} \)

The length of \( \overline{RT} \) is the vertical distance between \( R(1, 3) \) and \( T(1, 5) \). Using the distance formula for vertical line (\( x \)-coordinates same): \( |y_2 - y_1| = |5 - 3| = 2 \).

Step4: Length of \( \overline{ST} \)

Using Pythagorean theorem: \( ST = \sqrt{RS^2 + RT^2} = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 \) (or exact \( \sqrt{13} \), but approximate is ~3.6 or 3.61).

Answer:

• Point \( R \) coordinates: \( (1, 3) \)
• Length of \( \overline{RS} \): \( 3 \)
• Length of \( \overline{RT} \): \( 2 \)
• Length of \( \overline{ST} \): \( \approx 3.61 \) (or \( \sqrt{13} \))

(Note: If the drop-down options have specific values, adjust accordingly. For example, if options for \( R \) are like (1,3), (-2,5) etc., (1,3) is correct. For \( RS \), 3; \( RT \), 2; \( ST \), \( \sqrt{13} \approx 3.6 \) or 3.61.)