Question

visualizing the box method for the parallelogram
a parallelogram has coordinates of (5, 17), (10, 20), (18, 9), and (13, 6). which right triangle represents one of the cutouts from the box method?
four right triangle images, with side lengths: first (5,17), second (5,8), third (8,11), fourth (3,8)

Explanation:

Step1: Find the differences in coordinates

To find the legs of the right triangle, we calculate the differences in the x - coordinates and y - coordinates of two adjacent vertices of the parallelogram. Let's take two adjacent vertices, say \((5,17)\) and \((10,20)\), and another pair \((10,20)\) and \((18,9)\).

For the x - differences: Between \((5,17)\) and \((10,20)\), \(\Delta x=10 - 5=5\); between \((10,20)\) and \((18,9)\), \(\Delta x = 18-10 = 8\).

For the y - differences: Between \((5,17)\) and \((10,20)\), \(\Delta y=20 - 17 = 3\); between \((10,20)\) and \((18,9)\), \(\Delta y=9 - 20=- 11\) (we take the absolute value, so \(| - 11|=11\))? Wait, no, let's take \((5,17)\) and \((13,6)\). \(\Delta x=13 - 5 = 8\), \(\Delta y=6 - 17=-11\) (absolute value \(11\))? Wait, maybe a better approach: Let's consider the vectors. The sides of the parallelogram can be represented by vectors. Let's take two adjacent vertices, say \(A=(5,17)\), \(B=(10,20)\), \(C=(18,9)\), \(D=(13,6)\).

Vector \(\overrightarrow{AB}=(10 - 5,20 - 17)=(5,3)\)

Vector \(\overrightarrow{AD}=(13 - 5,6 - 17)=(8,- 11)\)

But when we use the box method for the parallelogram, the right triangles formed will have legs equal to the differences in x and y coordinates of the vertices. Let's calculate the differences between \((5,17)\) and \((18,9)\): \(\Delta x=18 - 5 = 13\)? No, maybe we should find the horizontal and vertical distances.

Wait, let's find the base and height - like differences. Let's take two points: \((5,17)\) and \((13,6)\):

The difference in x - coordinates: \(13 - 5=8\)

The difference in y - coordinates: \(6 - 17=- 11\), absolute value is \(11\). Wait, no, another pair: \((5,17)\) and \((10,20)\):

Difference in x: \(10 - 5 = 5\)

Difference in y: \(20 - 17=3\)

Wait, maybe I made a mistake. Let's calculate the length of the legs of the right triangles in the options. The options have right triangles with legs (5,17), (5,8), (8,11), (3,8).

Wait, let's find the slope between two points to see the relationship. The slope of the side between \((5,17)\) and \((10,20)\) is \(m=\frac{20 - 17}{10 - 5}=\frac{3}{5}\). The slope of the side between \((10,20)\) and \((18,9)\) is \(m=\frac{9 - 20}{18 - 10}=\frac{- 11}{8}\).

When we use the box method, we enclose the parallelogram in a rectangle. The right triangles cut out will have legs equal to the horizontal and vertical differences between the vertices. Let's find the horizontal and vertical extents.

The minimum x - coordinate is 5, maximum x - coordinate is 18, so the horizontal length of the box is \(18 - 5 = 13\)? No, that's not right. Wait, let's take two adjacent vertices, say \((5,17)\) and \((13,6)\):

\(\Delta x=13 - 5 = 8\), \(\Delta y=6 - 17=-11\) (so the vertical leg is 11, horizontal leg is 8). Another pair: \((10,20)\) and \((18,9)\): \(\Delta x=18 - 10 = 8\), \(\Delta y=9 - 20=-11\). And for \((5,17)\) and \((10,20)\): \(\Delta x = 5\), \(\Delta y=3\); \((13,6)\) and \((18,9)\): \(\Delta x=5\), \(\Delta y = 3\).

Wait, the right triangle with legs 3 and 8? Wait, no, the fourth option has legs 3 and 8? Wait, the fourth triangle has legs 3 and 8? Wait, the options:

First triangle: legs 5 and 17

Second: 5 and 8

Third: 8 and 11

Fourth: 3 and 8

Wait, let's calculate the differences between \((5,17)\) and \((10,20)\): \(\Delta x = 5\), \(\Delta y=3\). Between \((10,20)\) and \((18,9)\): \(\Delta x=8\), \(\Delta y=-11\). Wait, maybe I messed up. Let's calculate the length of the legs of the right triangle that should be cut out.

The vector between \((5,17)\) and \((10,20)\) is \((5,3)\), and between \((10,20)\) and \((18,9)\) is \((8,-11)\). When we form the box, the right triangles will have legs equal to the components of these vectors. Wait, the triangle with legs 3 and 8? No, wait the fourth triangle has legs 3 and 8? Wait, no, the fourth triangle has a horizontal leg of 8 and vertical leg of 3. Let's check the coordinates:

Take the point \((5,17)\), \((10,20)\), \((18,9)\), \((13,6)\). Let's find the difference between \((5,17)\) and \((13,6)\): \(x\) - difference: \(13 - 5 = 8\), \(y\) - difference: \(6 - 17=-11\) (no). Between \((10,20)\) and \((18,9)\): \(x\) - difference: \(18 - 10 = 8\), \(y\) - difference: \(9 - 20=-11\) (so absolute values 8 and 11). Wait, the third triangle has legs 8 and 11. Let's check the slope of the hypotenuse of the third triangle. The slope of the hypotenuse would be \(\frac{11}{8}\) (since it's a right triangle with legs 8 and 11). The slope of the side between \((10,20)\) and \((18,9)\) is \(\frac{9 - 20}{18 - 10}=\frac{-11}{8}\), which has the same absolute value of slope as \(\frac{11}{8}\), meaning that the right triangle with legs 8 and 11 is similar (or congruent in terms of leg lengths) to the triangle formed by the side of the parallelogram and the box. Wait, no, the right triangle cut out should have legs equal to the horizontal and vertical differences. Let's calculate the horizontal and vertical differences for the parallelogram's sides.

For the side from \((10,20)\) to \((18,9)\):

Horizontal change: \(18 - 10 = 8\)

Vertical change: \(9 - 20=-11\) (so the length of the vertical leg is 11, horizontal leg is 8)

So the right triangle with legs 8 and 11 (the third option) is the one that represents the cutout. Wait, but let's check another side. From \((5,17)\) to \((13,6)\):

Horizontal change: \(13 - 5 = 8\)

Vertical change: \(6 - 17=-11\)

Yes, so the right triangle with legs 8 and 11 is the cutout.

Step2: Match with the options

The third triangle in the options has legs 8 and 11, which matches the horizontal and vertical differences we calculated from the coordinates of the parallelogram's vertices.

Answer:

The right triangle with legs 8 and 11 (the third triangle in the given options)