To address this problem, we need to analyze the given parallelogram defined by the points [tex]\((5,17)\)[/tex], [tex]\((10,20)\)[/tex], [tex]\((18,9)\)[/tex], and [tex]\((13,6)\)[/tex]. The right triangle in question will be derived from these points. We will start by calculating the key distances between these points which will help us identify sides and diagonals of the parallelogram.
1. Calculate the length of the sides of the parallelogram:
- Distance between [tex]\((5,17)\)[/tex] and [tex]\((10,20)\)[/tex]:
[tex]\[
d_1 = \sqrt{(10-5)^2 + (20-17)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83
\][/tex]
- Distance between [tex]\((10,20)\)[/tex] and [tex]\((18,9)\)[/tex]:
[tex]\[
d_2 = \sqrt{(18-10)^2 + (9-20)^2} = \sqrt{64 + 121} = \sqrt{185} \approx 13.60
\][/tex]
- Distance between [tex]\((18,9)\)[/tex] and [tex]\((13,6)\)[/tex]:
[tex]\[
d_3 = \sqrt{(13-18)^2 + (6-9)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83
\][/tex]
- Distance between [tex]\((13,6)\)[/tex] and [tex]\((5,17)\)[/tex]:
[tex]\[
d_4 = \sqrt{(5-13)^2 + (17-6)^2} = \sqrt{64 + 121} = \sqrt{185} \approx 13.60
\][/tex]
2. Calculate the length of the diagonals of the parallelogram:
- Distance between [tex]\((5,17)\)[/tex] and [tex]\((18,9)\)[/tex]:
[tex]\[
d_5 = \sqrt{(18-5)^2 + (9-17)^2} = \sqrt{169 + 64} = \sqrt{233} \approx 15.26
\][/tex]
- Distance between [tex]\((10,20)\)[/tex] and [tex]\((13,6)\)[/tex]:
[tex]\[
d_6 = \sqrt{(13-10)^2 + (6-20)^2} = \sqrt{9 + 196} = \sqrt{205} \approx 14.32
\][/tex]
3. Identify the right triangle:
A right triangle in the context of cutting a parallelogram would likely be formed by one of its diagonals and two connecting sides meeting at a vertex. Let's use the lengths just computed to frame the problem:
- For a right triangle, we could consider one diagonal and the two adjacent connected sides. Let’s take the diagonal [tex]\((5,17) \rightarrow (18,9)\)[/tex] and its two adjacent sides:
[tex]\[
\text{Sides } \approx 13.60 \text{ units } \text{(connecting 5,17 to 10,20 and 10,20 to 18,9)}
\][/tex]
- If we look at the vertices’ distances carefully, it can help us check the right triangle created by these dimensions.
Based on the calculated distances and geometry, the right triangle is formed by the sides [tex]\(\approx 13.60\)[/tex] units connecting points [tex]\((5, 17)\)[/tex], [tex]\((10, 20)\)[/tex], and [tex]\((18, 9)\)[/tex]. This confirms that one of the right triangles from the box method could be visualized and characterized accurately by examining side lengths and right angles within the parallelogram.
