Question

jacob drew a blueprint of the deck that he is going to build in his backyard, as shown by trapezoid abcd on the coordinate grid below. each unit on the coordinate grid represents 1 yard. what is the perimeter, in yards, of jacobs deck? p = \square yards

Explanation:

Step1: Identify Coordinates

First, find the coordinates of each vertex:
- \( D(-6, 0) \)
- \( A(-2, 4) \)
- \( B(-1, 4) \)
- \( C(-1, 0) \)

Step2: Calculate Side Lengths

- \( DC \): Distance from \( D(-6,0) \) to \( C(-1,0) \). Since y-coordinates are same, length is \( |-1 - (-6)| = 5 \) yards.
- \( CB \): Distance from \( C(-1,0) \) to \( B(-1,4) \). x-coordinates same, length is \( |4 - 0| = 4 \) yards.
- \( AB \): Distance from \( A(-2,4) \) to \( B(-1,4) \). y-coordinates same, length is \( |-1 - (-2)| = 1 \) yard.
- \( AD \): Distance from \( A(-2,4) \) to \( D(-6,0) \). Use distance formula \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). So \( \sqrt{(-6 - (-2))^2 + (0 - 4)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \)? Wait, no, wait coordinates: Wait \( D(-6,0) \), \( A(-2,4) \). So \( x \)-difference: \( -2 - (-6) = 4 \), \( y \)-difference: \( 4 - 0 = 4 \). So distance is \( \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2} \approx 5.656 \)? Wait, but maybe I misread coordinates. Wait the graph: Let's recheck. Wait \( D \) is at (-6, 0)? Wait the grid: x-axis from -8 to 8, y-axis from -8 to 8. Wait \( D \) is at (-6, 0)? \( A \) at (-2, 4)? \( B \) at (-1, 4)? \( C \) at (-1, 0)? Wait no, maybe \( D \) is at (-5, 0)? Wait the original graph: Let's see, the x-axis has -6, -4, -2, 0, 2, etc. Wait the point \( D \) is at (-5, 0)? Wait maybe I misread. Wait the user's graph: Let's assume the coordinates are:

Wait \( D \) is at (-5, 0), \( A \) at (-2, 4), \( B \) at (-1, 4), \( C \) at (-1, 0). Wait no, the vertical line from \( C \) to \( B \): \( C \) is at (-1, 0), \( B \) at (-1, 4)? Wait no, the y-axis: the grid has 0, 2, 4, 6. Wait \( B \) is at (-1, 4)? \( A \) at (-2, 4)? Then \( AB \) is 1 unit (from x=-2 to x=-1, y=4). \( DC \): from \( D(-5, 0) \) to \( C(-1, 0) \): that's 4 units? Wait no, maybe the coordinates are:

Wait let's re-express:

- \( D \): (-5, 0)
- \( C \): (-1, 0) → so \( DC \) length: \( |-1 - (-5)| = 4 \) yards.
- \( C \): (-1, 0) to \( B \): (-1, 4) → length 4 yards (vertical).
- \( B \): (-1, 4) to \( A \): (-2, 4) → length 1 yard (horizontal).
- \( A \): (-2, 4) to \( D \): (-5, 0) → distance: \( \sqrt{(-5 - (-2))^2 + (0 - 4)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \) yards. Ah! That makes sense. So I misread \( D \)'s x-coordinate. So \( D \) is at (-5, 0), \( A \) at (-2, 4), \( B \) at (-1, 4), \( C \) at (-1, 0).

So recalculating:

- \( DC \): from (-5,0) to (-1,0): length \( 4 \) yards (since \( -1 - (-5) = 4 \)).
- \( CB \): from (-1,0) to (-1,4): length \( 4 \) yards (vertical, 4 - 0 = 4).
- \( AB \): from (-2,4) to (-1,4): length \( 1 \) yard (horizontal, -1 - (-2) = 1).
- \( AD \): from (-2,4) to (-5,0): distance \( \sqrt{(-5 - (-2))^2 + (0 - 4)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \) yards.

Now perimeter is sum of all sides: \( DC + CB + AB + AD = 4 + 4 + 1 + 5 = 14 \)? Wait no, wait trapezoid has four sides: \( DC \), \( CB \), \( BA \), \( AD \). Wait \( DC = 4 \), \( CB = 4 \), \( BA = 1 \), \( AD = 5 \). So total perimeter: 4 + 4 + 1 + 5 = 14? Wait but maybe my coordinate reading is wrong. Let's check again.

Alternative approach: Let's find the coordinates correctly.

From the graph:

- \( D \): (-5, 0) (since it's on x-axis, left of -4)
- \( C \): (-1, 0) (on x-axis, at x=-1)
- \( B \): (-1, 4) (vertical line from C, up to y=4)
- \( A \): (-2, 4) (horizontal line left from B to x=-2)

So:

- \( DC \): distance between D(-5,0) and C(-1,0): \( |-1 - (-5)| = 4 \)
- \( CB \): distance between C(-1,0) and B(-1,4): \( |4 - 0| = 4 \)
- \( BA \): distance between B(-1,4) and A(-2,4): \( |-2 - (-1)| = 1 \)
- \( AD \): distance between A(-2,4) and D(-5,0): \( \sqrt{(-5 - (-2))^2 + (0 - 4)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Now sum all sides: \( 4 + 4 + 1 + 5 = 14 \). Wait but maybe the coordinates are different. Wait another way: Maybe \( D \) is at (-6, 0), \( C \) at (-1, 0), so \( DC = 5 \), \( CB = 4 \), \( BA = 1 \), \( AD = \sqrt{( -6 + 2)^2 + (0 - 4)^2} = \sqrt{(-4)^2 + (-4)^2} = \sqrt{32} \approx 5.656 \), then perimeter would be 5 + 4 + 1 + 5.656 ≈ 15.656, but that doesn't make sense. So likely my first coordinate reading was wrong, and \( D \) is at (-5, 0), making \( AD = 5 \), so perimeter 14. Wait but maybe the correct coordinates are:

Wait the grid: each unit is 1 yard. Let's count the squares.

- \( DC \): from D to C: how many units? From x=-5 to x=-1: 4 units (since -1 - (-5) = 4)
- \( CB \): from C to B: 4 units (from y=0 to y=4)
- \( AB \): from B to A: 1 unit (from x=-1 to x=-2)
- \( AD \): from A to D: let's count the horizontal and vertical differences. From A(-2,4) to D(-5,0): horizontal difference is -3 (from -2 to -5), vertical difference is -4 (from 4 to 0). So distance is \( \sqrt{(-3)^2 + (-4)^2} = 5 \), which is a 3-4-5 right triangle. So that's correct.

Thus, perimeter is \( DC + CB + AB + AD = 4 + 4 + 1 + 5 = 14 \) yards. Wait but maybe I made a mistake. Wait trapezoid has two parallel sides: \( AB \) and \( DC \) are horizontal (both y=4 and y=0), so they are parallel. So \( AB = 1 \), \( DC = 4 \), height is 4 (vertical distance between y=0 and y=4). Then the non-parallel sides: \( CB = 4 \) (vertical) and \( AD = 5 \) (the slant side). So sum: 1 + 4 + 4 + 5 = 14.

Answer:

\boxed{14}