Question

silvio wants to find the perimeter of quadrilateral abcd. he begins by computing the length of side \\(\overline{ab}\\) using the distance formula. finish silvios work to find the length of side \\(\overline{ab}\\). silvio work \\(ab = \sqrt{(-3 - 1)^2 + (2 - 5)^2}\\) \\(= \sqrt{(-4)^2 + (-3)^2}\\) what is the perimeter of quadrilateral abcd? 17 units 18 units 15 units 20 units

Explanation:

Part 1: Find length of \( \overline{AB} \)

Step1: Calculate squares

First, calculate \( (-4)^2 \) and \( (-3)^2 \). \( (-4)^2 = 16 \), \( (-3)^2 = 9 \).
So, \( AB=\sqrt{16 + 9} \)

Step2: Sum and square root

Sum the values inside the square root: \( 16+9 = 25 \). Then take the square root: \( \sqrt{25}=5 \). So length of \( \overline{AB} \) is 5 units.

Part 2: Find perimeter of quadrilateral \( ABCD \)

From the graph, we can observe the other sides:

• \( \overline{BC} \): Vertical side, from \( (1,5) \) to \( (1,-1) \) (assuming the bottom vertex is \( (1,-1) \)), length is \( 5 - (-1)=6 \)? Wait, no, looking at the grid, \( A(-3,2) \), \( B(1,5) \), then the vertical side from \( B(1,5) \) down to the bottom: let's check the coordinates. Wait, the bottom vertices: let's see, the left vertical side is from \( A(-3,2) \) down to \( (-3,-1) \)? Wait, maybe better to count grid units.

Wait, \( A(-3,2) \), the left vertical segment: from \( (-3,2) \) to \( (-3,-1) \): length is \( 2 - (-1)=3 \)? No, wait the graph: \( A(-3,2) \), then down to \( (-3,-1) \) (the bottom left), then right to \( (1,-1) \) (bottom right), then up to \( B(1,5) \)? Wait no, \( B \) is \( (1,5) \), then the vertical side from \( B(1,5) \) to \( (1,-1) \): that's length \( 5 - (-1)=6 \)? Wait, no, let's re - examine:

Wait, \( A(-3,2) \), \( B(1,5) \) (length 5 as found). Then \( \overline{BC} \): from \( (1,5) \) down to \( (1,-1) \): the y - coordinates change from 5 to - 1, so length is \( 5-(-1)=6 \)? No, wait the grid: each square is 1 unit. From \( (1,5) \) down to \( (1, - 1) \): the number of units is \( 5 - (-1)=6 \)? Wait, no, \( 5-(-1)=6 \), but let's check the horizontal side: from \( (-3,-1) \) to \( (1,-1) \): the x - coordinates change from - 3 to 1, so length is \( 1-(-3)=4 \). And the left vertical side: from \( (-3,2) \) to \( (-3,-1) \): length is \( 2-(-1)=3 \)? Wait, this is confusing. Wait, maybe the quadrilateral is a rectangle? No, \( AB \) is length 5, \( BC \): let's count the grid. From \( B(1,5) \) down to \( (1, - 1) \): that's 6 units? Wait, no, maybe I made a mistake. Wait, the coordinates: \( A(-3,2) \), \( B(1,5) \), \( C(1,-1) \), \( D(-3,-1) \).

So:

• \( AB \): length 5 (as calculated)
• \( BC \): from \( (1,5) \) to \( (1,-1) \): \( \vert5 - (-1)\vert=6 \)
• \( CD \): from \( (1,-1) \) to \( (-3,-1) \): \( \vert1-(-3)\vert = 4 \)
• \( DA \): from \( (-3,-1) \) to \( (-3,2) \): \( \vert2-(-1)\vert=3 \)

Wait, no, that can't be. Wait, maybe the quadrilateral is a trapezoid? Wait, no, looking at the graph, \( A(-3,2) \), \( B(1,5) \), then the vertical line from \( B(1,5) \) down to \( (1, - 1) \), then horizontal to \( (-3,-1) \), then vertical up to \( A(-3,2) \).

So sides:

• \( AB \): 5 (calculated)
• \( BC \): from \( (1,5) \) to \( (1,-1) \): length is \( 5-(-1)=6 \)
• \( CD \): from \( (1,-1) \) to \( (-3,-1) \): length is \( 1 - (-3)=4 \)
• \( DA \): from \( (-3,-1) \) to \( (-3,2) \): length is \( 2-(-1)=3 \)

Now perimeter \( =AB + BC+CD + DA=5 + 6+4 + 3=18 \) units.

Answer:

for length of \( \overline{AB} \): 5 units