Question

graph the figure with the given vertices in the coordinate plane. find the perimeter. l(0,2), m(3,6), n(6,6), p(6,2) ... choose the correct graph on the right. the perimeter of lmnp is \boxed{} units.

Explanation:

Step1: Identify the shape and side lengths

First, we analyze the coordinates:

• \( LM \): Distance between \( L(0,2) \) and \( M(3,6) \). Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), we get \( \sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
• \( MN \): Distance between \( M(3,6) \) and \( N(6,6) \). Since \( y \)-coordinates are the same, it's \( |6 - 3| = 3 \).
• \( NP \): Distance between \( N(6,6) \) and \( P(6,2) \). Since \( x \)-coordinates are the same, it's \( |6 - 2| = 4 \).
• \( PL \): Distance between \( P(6,2) \) and \( L(0,2) \). Since \( y \)-coordinates are the same, it's \( |6 - 0| = 6 \). Wait, no, actually, let's re - check. Wait, the figure: Let's plot the points. \( L(0,2) \), \( M(3,6) \), \( N(6,6) \), \( P(6,2) \). So \( LM \): from (0,2) to (3,6), \( MN \): from (3,6) to (6,6) (length 3), \( NP \): from (6,6) to (6,2) (length 4), \( PL \): from (6,2) to (0,2) (length 6). Wait, but also, maybe we can see the shape. Alternatively, maybe it's a trapezoid. Wait, but let's calculate the perimeter as the sum of all sides. Wait, no, maybe I made a mistake. Wait, let's recalculate \( LM \): \( x \) difference is 3, \( y \) difference is 4, so length 5. \( MN \): 3 units (horizontal). \( NP \): 4 units (vertical). \( PL \): 6 units (horizontal). Wait, but that can't be. Wait, no, the coordinates: \( L(0,2) \), \( P(6,2) \): so \( PL \) is from \( x = 0 \) to \( x = 6 \), \( y = 2 \), so length 6. \( P(6,2) \) to \( N(6,6) \): \( y \) from 2 to 6, length 4. \( N(6,6) \) to \( M(3,6) \): \( x \) from 6 to 3, length 3. \( M(3,6) \) to \( L(0,2) \): distance \( \sqrt{(3 - 0)^2+(6 - 2)^2}=\sqrt{9 + 16}=5 \). So perimeter is \( 5 + 3 + 4 + 6=18 \)? Wait, no, that's not right. Wait, maybe the figure is a trapezoid with two parallel sides. Wait, \( MN \) is horizontal (from (3,6) to (6,6)), length 3. \( PL \) is horizontal (from (0,2) to (6,2)), length 6. \( NP \) is vertical (from (6,6) to (6,2)), length 4. \( LM \): from (0,2) to (3,6), length 5. Wait, but maybe I mis - identified the sides. Wait, the correct way is to list the sides in order: \( L \) to \( M \), \( M \) to \( N \), \( N \) to \( P \), \( P \) to \( L \). So \( LM = 5 \), \( MN = 3 \), \( NP = 4 \), \( PL = 6 \). Sum: \( 5+3 + 4+6 = 18 \). Wait, but let's check again. Wait, \( L(0,2) \), \( P(6,2) \): distance is \( 6 - 0=6 \) (since \( y \) is same). \( P(6,2) \), \( N(6,6) \): \( 6 - 2 = 4 \) (since \( x \) is same). \( N(6,6) \), \( M(3,6) \): \( 6 - 3=3 \) (since \( y \) is same). \( M(3,6) \), \( L(0,2) \): \( \sqrt{(3 - 0)^2+(6 - 2)^2}=\sqrt{9 + 16}=5 \). So perimeter is \( 5 + 3+4 + 6 = 18 \).

Step2: Sum the side lengths

Perimeter \( P=LM + MN+NP + PL \)
\( LM = 5 \), \( MN = 3 \), \( NP = 4 \), \( PL = 6 \)
\( P=5 + 3+4 + 6=18 \)

Answer:

18