Question

what is the perimeter of rectangle f g h i? perimeter = \boxed{} units

Explanation:

Step1: Find the length of the rectangle

Looking at the coordinates of points \( H(-6,9) \) and \( G(9,9) \), the length \( l \) is the difference in the x - coordinates. So \( l=\vert9 - (-6)\vert=15 \) units? Wait, no, wait the grid. Wait, let's check the coordinates again. Wait, point \( I \) is at \( (-6,5) \), \( F \) is at \( (9,5) \), \( H \) is at \( (-6,9) \), \( G \) is at \( (9,9) \). So the length of the horizontal side: from \( x=-6 \) to \( x = 9 \), the number of units is \( 9-(-6)=15 \)? Wait, no, wait the grid lines. Wait, each grid square is 1 unit. Let's count the horizontal distance between \( I(-6,5) \) and \( F(9,5) \). The x - coordinate of \( I \) is - 6, x - coordinate of \( F \) is 9. So the length \( l=9-(-6)=15 \)? Wait, no, wait maybe I made a mistake. Wait, let's check the vertical side. From \( I(-6,5) \) to \( H(-6,9) \), the y - coordinate difference is \( 9 - 5=4 \) units. So the width \( w = 4 \) units. Now the horizontal side: from \( -6 \) to \( 9 \), the number of units is \( 9-(-6)=15 \)? Wait, no, wait the distance between \( x=-6 \) and \( x = 9 \) is \( 9-(-6)=15 \)? Wait, but let's count the grid squares. From \( x=-6 \) to \( x = 0 \) is 6 units, from \( x = 0 \) to \( x = 9 \) is 9 units, so total \( 6 + 9=15 \) units. Wait, but the vertical side: from \( y = 5 \) to \( y = 9 \) is \( 9 - 5=4 \) units.

Step2: Use the perimeter formula for a rectangle

The formula for the perimeter \( P \) of a rectangle is \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. We found \( l = 15 \) units and \( w = 4 \) units. So \( P=2(15 + 4)=2\times19 = 38 \)? Wait, no, wait that can't be right. Wait, maybe I messed up the length. Wait, let's re - check the coordinates. Wait, point \( I \) is at \( (-6,5) \), \( F \) is at \( (9,5) \)? Wait, no, looking at the grid, the x - coordinate of \( I \) is - 6, and \( F \) is at \( 9 \)? Wait, no, maybe the x - coordinate of \( F \) is 9? Wait, the grid goes from - 10 to 10 on x. Wait, let's count the horizontal distance between \( I \) and \( F \). From \( x=-6 \) to \( x = 9 \), the number of units is \( 9-(-6)=15 \)? Wait, but maybe the length is 15 and width is 4. Wait, but let's check the vertical side: from \( I(-6,5) \) to \( H(-6,9) \), the y - difference is \( 9 - 5 = 4 \), so width is 4. Then length is the horizontal side: from \( -6 \) to \( 9 \), which is \( 9-(-6)=15 \). Then perimeter is \( 2\times(15 + 4)=2\times19 = 38 \). Wait, but maybe I made a mistake in the length. Wait, let's check the coordinates of \( G \) and \( H \). \( H \) is at \( (-6,9) \), \( G \) is at \( (9,9) \), so the distance between \( H \) and \( G \) is \( 9-(-6)=15 \), which is the length. And the distance between \( G \) and \( F \) is \( 9 - 5=4 \), which is the width. So perimeter \( P = 2\times(15 + 4)=38 \). Wait, but maybe the length is 14? Wait, no, let's count the grid squares. From \( x=-6 \) to \( x = 9 \), how many units? Let's list the x - coordinates: - 6, - 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Wait, from - 6 to 9, that's \( 9-(-6)=15 \) units (because the number of intervals between - 6 and 9 is \( 9-(-6)=15 \)). And the vertical side: from \( y = 5 \) to \( y = 9 \), that's \( 9 - 5 = 4 \) units. So perimeter is \( 2\times(15 + 4)=38 \).

Wait, maybe I made a mistake. Wait, let's check again. Let's take the coordinates:

- \( I(-6,5) \)
- \( F(9,5) \)
- \( G(9,9) \)
- \( H(-6,9) \)

So the length of \( IF \): distance between \( (-6,5) \) and \( (9,5) \) is \( \sqrt{(9 - (-6))^{2}+(5 - 5)^{2}}=\vert9+6\vert = 15 \)

The length of \( FG \): distance between \( (9,5) \) and \( (9,9) \) is \( \sqrt{(9 - 9)^{2}+(9 - 5)^{2}}=\vert9 - 5\vert=4 \)

Perimeter of rectangle is \( 2\times(15 + 4)=2\times19 = 38 \)

Answer:

38