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Galina is finding the area of triangle RST. To do so, she follows these steps:

[tex]\[
\begin{tabular}{|l|l|}
\hline
Step 1 & Draw a rectangle around triangle RST. \\
\hline
Step 2 & Find the area of the rectangle. \\
\hline
Step 3 & Find the area of the three right triangles created by the sides of the rectangle and the intersection with each vertex of the triangle. \\
\hline
Step 4 & Subtract the area of the right triangles from the area of the rectangle. \\
\hline
\end{tabular}
\][/tex]

To solve the problem using these steps, what are the dimensions of the rectangle she should draw?



Answer :

GinnyAnswer
Let's break down the solution step by step to determine the dimensions of the rectangle that Galina should draw around triangle RST.

1. Identify the coordinates of triangle RST:
- The vertices of triangle RST have the following coordinates:
- [tex]\( R(x1, y1) = (2, 3) \)[/tex]
- [tex]\( S(x2, y2) = (5, 7) \)[/tex]
- [tex]\( T(x3, y3) = (9, 1) \)[/tex]

2. Determine the smallest rectangle that can include the entire triangle:

1. Find the minimum and maximum x and y coordinates among the vertices:
- Minimum x-coordinate ([tex]\( \text{min\_x} \)[/tex]): The smallest x-coordinate is 2.
- Maximum x-coordinate ([tex]\( \text{max\_x} \)[/tex]): The largest x-coordinate is 9.
- Minimum y-coordinate ([tex]\( \text{min\_y} \)[/tex]): The smallest y-coordinate is 1.
- Maximum y-coordinate ([tex]\( \text{max\_y} \)[/tex]): The largest y-coordinate is 7.

2. Calculate the width and height of the rectangle:
- Width ([tex]\( \text{width} \)[/tex]) of the rectangle is the difference between the maximum and minimum x-coordinates:
[tex]\[ \text{width} = \text{max\_x} - \text{min\_x} = 9 - 2 = 7 \][/tex]
- Height ([tex]\( \text{height} \)[/tex]) of the rectangle is the difference between the maximum and minimum y-coordinates:
[tex]\[ \text{height} = \text{max\_y} - \text{min\_y} = 7 - 1 = 6 \][/tex]

3. Determine the coordinates of the bottom-left and top-right corners of the rectangle:
- Bottom-left corner: ([tex]\( \text{bottom\_left\_x}, \text{bottom\_left\_y} \)[/tex]):
[tex]\[ (\text{min\_x}, \text{min\_y}) = (2, 1) \][/tex]
- Top-right corner: ([tex]\( \text{top\_right\_x}, \text{top\_right\_y} \)[/tex]):
[tex]\[ (\text{max\_x}, \text{max\_y}) = (9, 7) \][/tex]

Therefore, the dimensions of the rectangle that Galina should draw around triangle RST are:

- Width: 7 units
- Height: 6 units
- Bottom-left corner coordinates: (2, 1)
- Top-right corner coordinates: (9, 7)

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