Question

what is the perimeter of δfgh?
coordinate grid with triangle fgh
write your answer as an integer or as a decimal rounded to the nearest tenth.
perimeter = \boxed{} units

Explanation:

Step1: Identify coordinates of points

From the graph, we can determine the coordinates of the vertices of the triangle:
- \( F(-8, -4) \)
- \( G(7, -9) \) (Wait, looking at the grid, actually, let's re - check. Wait, the x - coordinate of G: looking at the grid, the vertical line for G is at x = 7? Wait, no, the grid lines: from the graph, F is at (-8, -4), G is at (7, -9)? Wait, no, maybe I misread. Wait, the x - axis: the points are on the grid. Let's look again. The horizontal (x) and vertical (y) axes. Let's find the coordinates properly:

Looking at the graph:
- Point F: x=-8, y = - 4 (since it's 8 units left on x - axis and 4 units down on y - axis)
- Point G: x = 7, y=-9? Wait, no, the vertical line for G: looking at the grid, the x - coordinate of G is 7? Wait, no, the grid has x from - 10 to 10, y from - 10 to 10. Wait, the point G is at (7, - 9)? Wait, no, maybe the coordinates are:

Wait, the vertical line for H: x = 7, y = 7 (since it's 7 units right on x - axis and 7 units up on y - axis). G is at (7, - 9) (7 units right, 9 units down). F is at (-8, - 4) (-8 units right, 4 units down).

Step2: Calculate length of FG

Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \)
For \( F(-8,-4) \) and \( G(7,-9) \)
\( x_1=-8,y_1 = - 4,x_2 = 7,y_2=-9 \)
\( FG=\sqrt{(7 - (-8))^2+(-9 - (-4))^2}=\sqrt{(15)^2+(-5)^2}=\sqrt{225 + 25}=\sqrt{250}\approx15.8 \)

Step3: Calculate length of GH

Since G and H have the same x - coordinate (x = 7), the distance is the difference in y - coordinates.
\( H(7,7) \), \( G(7,-9) \)
\( GH=\vert7-(-9)\vert=\vert16\vert = 16 \)

Step4: Calculate length of FH

Using distance formula for \( F(-8,-4) \) and \( H(7,7) \)
\( x_1=-8,y_1=-4,x_2 = 7,y_2 = 7 \)
\( FH=\sqrt{(7-(-8))^2+(7 - (-4))^2}=\sqrt{(15)^2+(11)^2}=\sqrt{225 + 121}=\sqrt{346}\approx18.6 \) Wait, no, wait, maybe I made a mistake in coordinates. Wait, let's re - examine the graph.

Wait, maybe the coordinates are:
- F: (-8, - 4)
- G: (7, - 9)? No, maybe the y - coordinate of H is 7, G is (7, - 9)? Wait, no, the vertical line for H and G: the x - coordinate is 7. The y - coordinate of H is 7 (since it's 7 units above x - axis), G is 9 units below x - axis, so y=-9. F is at (-8, - 4) (8 units left, 4 units down).

Wait, but maybe I messed up the coordinates. Let's try again. Let's look at the grid:

Each grid square is 1 unit. So:

- F: x=-8, y=-4 (because from the origin (0,0), move 8 left (x=-8) and 4 down (y = - 4))
- G: x = 7, y=-9 (7 right, 9 down)
- H: x = 7, y = 7 (7 right, 7 up)

Now, calculate GH: vertical distance between G(7,-9) and H(7,7). The formula for vertical distance is \( \vert y_2 - y_1\vert \), so \( \vert7-(-9)\vert=\vert16\vert = 16 \)

Calculate FG: distance between F(-8,-4) and G(7,-9). Using distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \)
\( x_2 - x_1=7-(-8)=15 \), \( y_2 - y_1=-9-(-4)=-5 \)
So \( FG=\sqrt{15^2+(-5)^2}=\sqrt{225 + 25}=\sqrt{250}\approx15.81 \)

Calculate FH: distance between F(-8,-4) and H(7,7)
\( x_2 - x_1=7-(-8)=15 \), \( y_2 - y_1=7-(-4)=11 \)
\( FH=\sqrt{15^2+11^2}=\sqrt{225 + 121}=\sqrt{346}\approx18.60 \)

Now, perimeter \( P=FG + GH+FH \)
\( P\approx15.81+16 + 18.60=50.41\approx50.4 \) Wait, that can't be right. Wait, maybe I misread the coordinates. Let's check again.

Wait, maybe the coordinates of G are (7, - 9)? No, maybe the y - coordinate of G is - 9? Wait, the grid: the bottom of the grid is y=-10. So G is at (7, - 9), H is at (7,7), F is at (-8, - 4).

Wait, another way: maybe the coordinates are:

F(-8, - 4), G(7, - 9), H(7,7)

Wait, but let's check the horizontal and vertical distances.

Wait, maybe I made a mistake in the x - coordinate of F. Let's look at the left - most point: F is at x=-8, y=-4. G is at x = 7, y=-9. H is at x = 7, y = 7.

Wait, let's recalculate FG:

\( \Delta x=7-(-8)=15 \), \( \Delta y=-9-(-4)=-5 \)
\( FG=\sqrt{15^2 + (-5)^2}=\sqrt{225 + 25}=\sqrt{250}\approx15.8 \)

GH: vertical line, so length is \( 7-(-9)=16 \) (since y goes from - 9 to 7, the difference is 16)

FH: \( \Delta x=7-(-8)=15 \), \( \Delta y=7-(-4)=11 \)
\( FH=\sqrt{15^2+11^2}=\sqrt{225 + 121}=\sqrt{346}\approx18.6 \)

Perimeter: \( 15.8+16 + 18.6=50.4 \)

Wait, but maybe the coordinates are different. Let's check the graph again. Wait, the x - axis: from - 10 to 10, y - axis from - 10 to 10. The point F is at (-8, - 4), G is at (7, - 9), H is at (7,7).

Alternatively, maybe the coordinates of G are (7, - 9), H is (7,7), F is (-8, - 4).

Wait, another approach: maybe the triangle is a right triangle? No, because GH is vertical, FH is a slant, FG is a slant.

Wait, maybe I made a mistake in the coordinates of F. Let's see, the left - hand side of the red marking: F is at x=-8, y=-4.

Wait, let's calculate the distance between F and G again. If F is (-8, - 4) and G is (7, - 9):

\( (x_2 - x_1)=7 - (-8)=15 \), \( (y_2 - y_1)=-9-(-4)=-5 \)

So \( FG=\sqrt{15^2+(-5)^2}=\sqrt{225 + 25}=\sqrt{250}\approx15.8 \)

Distance between G and H: G(7, - 9), H(7,7). Since x - coordinates are the same, distance is \( \vert7-(-9)\vert = 16 \)

Distance between F and H: F(-8, - 4), H(7,7)

\( (x_2 - x_1)=7-(-8)=15 \), \( (y_2 - y_1)=7-(-4)=11 \)

\( FH=\sqrt{15^2+11^2}=\sqrt{225 + 121}=\sqrt{346}\approx18.6 \)

Perimeter: \( 15.8+16 + 18.6 = 50.4 \)

Wait, but maybe the coordinates are F(-8, - 4), G(7, - 9), H(7,7). So the perimeter is approximately 50.4.

Answer:

\( 50.4 \)