Question

A rectangle is shown in the figure below. What is the perimeter? Units are in inches.

the points are

(3,4)

(7,2)

(-3,1)

(1,-1)

(1 point)
Responses

105–√ in
105–√ in

155–√ in
155–√ in

265–√ in
265–√ in

55–√ in

Answer 1

To find the perimeter of the rectangle defined by the points (3, 4), (7, 2), (-3, 1), and (1, -1), we first need to determine which pairs of points form the vertices of the rectangle.

1. **Find the lengths of the rectangle's sides:**
To do this, we can calculate the distances between the pairs of points. Let's find the lengths of segments formed by pairs of points.

- Length between (3, 4) and (7, 2):
\[
d = \sqrt{(7 - 3)^2 + (2 - 4)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

- Length between (3, 4) and (-3, 1):
\[
d = \sqrt{(-3 - 3)^2 + (1 - 4)^2} = \sqrt{-6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]

- Length between (3, 4) and (1, -1):
\[
d = \sqrt{(1 - 3)^2 + (-1 - 4)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}
\]

- Length between (7, 2) and (1, -1):
\[
d = \sqrt{(1 - 7)^2 + (-1 - 2)^2} = \sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]

- Length between (-3, 1) and (7, 2):
\[
d = \sqrt{(7 - (-3))^2 + (2 - 1)^2} = \sqrt{(10)^2 + (1)^2} = \sqrt{100 + 1} = \sqrt{101}
\]

- Length between (1, -1) and (1, -1):
(This point is repeated)

2. **Identify pairs forming the sides:**
We can see some distances are equal, indicating parallel sides.

3. **Find the perimeter:**
Let’s assume the pairs that form the rectangle are (3, 4) with (1, -1) and (7, 2) with (-3, 1) based on their coordinates.

- One side length can be \( 2\sqrt{5} \) (from previous calculations)
- The other length would be between (3, 4) and (7, 2) or (1, -1)

- Calculate the perimeter:
Since the perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) and \( w \) are lengths of the sides:
\[
P = 2(2\sqrt{5} + 3\sqrt{5})
\]
\[
P = 2(5\sqrt{5}) = 10\sqrt{5}
\]

The numerical choices given seem to follow a different understanding of rectangle properties. Please confirm if there's a misunderstanding about the application or sources of rectangle properties or clarify more about choices.

Based on the given choices:

- \( 105 - \sqrt{\text{in}}\)
- \( 155 - \sqrt{\text{in}}\)
- \( 265 - \sqrt{\text{in}}\)
- \( 55 - \sqrt{\text{in}}\)

None exactly match or seem compatible with conventional interpretation. Please check images/models you've based this on for specific boundaries.