Jared takes a rectangular piece of fabric and makes a diagonal cut from one corner to the opposite corner. The cut he makes is 15 centimeters long and the width of the fabric is 9 centimeters. What is the fabric's length?
First, let's calculate the total distance traveled by the Island Loop Ferry along its route. The ferry travels in sections of straight lines that form a right triangle.
The first leg of the trip is due south from Seabreeze Village to Oceanview, which is 8 miles long. The second leg is due east from Oceanview to Portlandville, which is 15 miles long. These two legs form the perpendicular sides of a right triangle.
Now, to calculate the length of the third leg (the hypotenuse of the triangle), which is the direct trip back from Portlandville to Seabreeze Village, we use the Pythagorean theorem:
a^2 + b^2 = c^2
Where:
a = 8 miles (southward trip)
b = 15 miles (eastward trip)
c = the hypotenuse (the direct trip back)
Substitute the values and solve for c:
8^2 + 15^2 = c^2
64 + 225 = c^2
289 = c^2
Now take the square root of both sides:
c = √289
c = 17 miles
So, the ferry travels 8 miles south, then 15 miles east, and finally 17 miles diagonally back to Seabreeze Village. The total distance traveled for one complete route is the sum of all three legs:
Total distance = 8 + 15 + 17 = 40 miles
The ferry uses 1 gallon of fuel to travel 5 miles, so the fuel consumption for the entire route can be calculated by dividing the total distance by the distance traveled per gallon of fuel:
Fuel used = Total distance / Distance per gallon
Fuel used = 40 miles / 5 miles per gallon
Fuel used = 8 gallons
Therefore, the Island Loop Ferry uses 8 gallons of fuel to complete the route one time. There's no need to round since we have an exact answer.
When Jared makes a diagonal cut from one corner of the rectangular fabric to the opposite corner, he essentially creates two right-angled triangles. The diagonal cut represents the hypotenuse of these triangles, the width of 9 centimeters is one side (let's call it "a") and the length we're trying to determine is the other side (let's call it "b").
We can use the Pythagorean theorem to find the length of the rectangle since the triangle formed is a right triangle. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
So we have:
c^2 = a^2 + b^2
where c is the diagonal (15 cm), a is the width (9 cm), and b is the unknown length we want to find.
Now we plug in the values and solve for b:
15^2 = 9^2 + b^2
225 = 81 + b^2
b^2 = 225 - 81
b^2 = 144
Now we find the square root of both sides to solve for b:
b = √144
b = 12
Therefore, the length of the fabric is 12 centimeters.
Two cars leave the same parking lot, with one heading north and the other heading east. After several minutes, the eastbound car has traveled 3 miles. If the two cars are now a straight-line distance of 5 miles apart, how far has the northbound car traveled?
This problem can again be solved using the Pythagorean theorem. In this case, the cars are moving at right angles to each other— one goes east while the other goes north. After they've traveled for a while, the straight-line distance between the two cars will be the hypotenuse of the right triangle formed by their paths.
Let's define the variables as follows:
- The eastbound car has traveled 3 miles, which is one leg of the right triangle (let's call it "a").
- The northbound car has traveled an unknown distance, which is the other leg of the triangle (let's call it "b").
- The straight-line distance between the two cars is the hypotenuse of the right triangle and is given as 5 miles (let's call it "c").
The Pythagorean theorem states that:
a^2 + b^2 = c^2
We know that:
a = 3 miles
c = 5 miles
We can plug these values into the theorem and solve for b:
3^2 + b^2 = 5^2
9 + b^2 = 25
b^2 = 25 - 9
b^2 = 16
Take the square root of both sides to find b:
b = √16
b = 4
The northbound car has traveled 4 miles.
Hannah's bedroom is rectangular. The length of one wall of Hannah's bedroom is 16 feet. The length from one corner of the bedroom to the diagonally opposite corner is 20 feet. What is the length of the other wall?
Using the Pythagorean theorem again, we can solve for the length of the other wall in Hannah's bedroom. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
Where:
c = the diagonal across the room (20 feet),
a = the length of one wall (16 feet),
b = the length of the other wall, which we want to find.
The equation based on the theorem is:
a^2 + b^2 = c^2
Plugging the known values into the equation we have:
16^2 + b^2 = 20^2
256 + b^2 = 400
Now solve for b^2:
b^2 = 400 - 256
b^2 = 144
Take the square root of both sides to find b:
b = √144
b = 12
Therefore, the length of the other wall is 12 feet.
Dan is building a skateboarding ramp by propping the end of a piece of wood on a cinder block. If the ramp starts 72 centimeters from the cinder block and the piece of wood is 90 centimeters long, how tall is the cinder block?
Dan has effectively created a right-angled triangle with his skateboarding ramp, where the piece of wood acts as the hypotenuse and is 90 centimeters long. The distance from the ramp's base to the cinder block is one leg of the triangle and measures 72 centimeters. We need to find the height of the cinder block, which represents the other leg of the triangle.
We can use the Pythagorean theorem, which relates the sides of a right-angled triangle as follows:
a^2 + b^2 = c^2
In this scenario:
a = the height of the cinder block (which we are trying to find),
b = the horizontal distance from the ramp's base to the cinder block (72 cm),
c = the length of the ramp, i.e., the hypotenuse (90 cm).
Substituting the known values into the Pythagorean theorem, we get:
a^2 + 72^2 = 90^2
a^2 + 5184 = 8100
Now we solve for a^2:
a^2 = 8100 - 5184
a^2 = 2916
Finally, we find the value of a by taking the square root of both sides:
a = √2916
a = 54
Therefore, the height of the cinder block, which is the vertical distance from the ground to the top of the ramp where the wood is resting, is 54 centimeters.
Two airplanes leave the same airport. One heads north, and the other heads east. After some time, the northbound airplane has traveled 21 miles, and the eastbound airplane has traveled 72 miles. How far apart are the two airplanes?
The two airplanes are creating a right-angled triangle with their paths. The northbound airplane's path represents one leg of the triangle, and the eastbound airplane's path represents the other leg. The straight-line distance between the airplanes will be the hypotenuse of the right triangle.
Let's denote:
- The distance traveled by the northbound airplane as "a" equals 21 miles
- The distance traveled by the eastbound airplane as "b" equals 72 miles
- The straight-line distance between the airplanes as "c"
Using the Pythagorean theorem:
a^2 + b^2 = c^2
Plugging in the values for a and b, we can solve for c:
21^2 + 72^2 = c^2
441 + 5184 = c^2
5625 = c^2
Now, we find the square root of 5625 to solve for c:
c = √5625
c = 75
Thus, the two airplanes are 75 miles apart.
Cora takes a sheet of paper and cuts from one corner to the opposite corner, making two triangles. If the piece of paper is 24 inches long and 18 inches wide, how long is the diagonal cut that Cora made
Cora's cut across the sheet of paper creates two right triangles, with the diagonal cut serving as the hypotenuse for both triangles. The length of the paper is one side of the triangles, and the width is the other side.
Given the length (a) is 24 inches and the width (b) is 18 inches, we can use the Pythagorean theorem to find the length of the diagonal cut (c).
The Pythagorean theorem states that for a right triangle:
a^2 + b^2 = c^2
Plugging in the values we have:
24^2 + 18^2 = c^2
576 + 324 = c^2
900 = c^2
To find the length of the diagonal cut (c), we take the square root of 900:
c = √900
c = 30
Therefore, the diagonal cut that Cora made is 30 inches long.
There is a rope running from the top of a flagpole to a hook in the ground. The flagpole is 12 meters high, and the hook is 9 meters from its base. How long is the rope?
The rope, the flagpole, and the distance from the hook to the base of the flagpole form a right-angled triangle.
We can use the Pythagorean theorem to solve for the length of the rope. The theorem states:
a^2 + b^2 = c^2
Where:
a = the height of the flagpole (12 meters),
b = the distance from the hook to the base of the flagpole (9 meters),
c = the length of the rope (which we're trying to find).
Let's plug in the values we know:
12^2 + 9^2 = c^2
144 + 81 = c^2
225 = c^2
We take the square root of 225 to find the length of the rope:
c = √225
c = 15
The rope is 15 meters long.
Frank is deep-sea diving with two friends. Zane is exploring a coral reef 7 meters in front of Frank, and Valeria is floating on the surface directly above Frank. If Valeria and Zane are 10 meters apart, how far apart are Frank and Valeria? If necessary, round to the nearest tenth.
To find how far apart Frank and Valeria are, we can imagine the scenario as creating a right-angled triangle. Here, Zane's position relative to Frank's is one leg of the triangle (7 meters in front of Frank), and the distance from Frank to Valeria forms the other leg, which is what we're trying to find. The distance between Valeria and Zane, which is 10 meters, is the hypotenuse of the triangle.
We can use the Pythagorean theorem to find the distance from Frank to Valeria (the other leg of the triangle), let's call it "a":
a^2 + b^2 = c^2
Where:
a = distance from Frank to Valeria (unknown),
b = distance from Zane to Frank (7 meters),
c = distance from Valeria to Zane (10 meters).
Plugging in the values we have:
a^2 + 7^2 = 10^2
a^2 + 49 = 100
Now we subtract 49 from both sides to solve for a^2:
a^2 = 100 - 49
a^2 = 51
Finally, we take the square root of both sides to find "a":
a ≈ √51
a ≈ 7.1 meters
So, rounding to the nearest tenth, Frank and Valeria are approximately 7.1 meters apart.
three ballet dancers are positioned on stage. If Roxanne is 7 meters straight behind Jen and 7 meters directly left of Ann, how far is Jen from Ann? If necessary, round to the nearest tenth.
The setup with the three ballet dancers forms a right-angled triangle on the stage. Roxanne is positioned at the right angle of the triangle, 7 meters straight behind Jen (one leg of the triangle) and 7 meters directly left of Ann (the other leg of the triangle).
To find the distance between Jen and Ann, which is the hypotenuse of the right triangle, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
where:
a = the distance from Roxanne to Jen (7 meters),
b = the distance from Roxanne to Ann (7 meters),
c = the distance from Jen to Ann (what we're trying to find).
Both a and b are 7 meters, so we have:
7^2 + 7^2 = c^2
49 + 49 = c^2
98 = c^2
Now to solve for c, which is the distance from Jen to Ann, we take the square root of both sides:
c ≈ √98
c ≈ 9.9 meters
Rounded to the nearest tenth, Jen is approximately 9.9 meters from Ann.
Toby is building a slide for his kids. The ladder is 2 meters tall and the slide is 4 meters long. What is the distance between the ladder and the bottom of the slide? If necessary, round to the nearest tenth
Toby's slide and ladder setup creates a right triangle. The ladder is one leg of the triangle (height), the slide is the hypotenuse, and the distance from the ladder to the bottom of the slide is the other leg (base).
Given:
Height (a) = 2 meters
Hypotenuse (c) = 4 meters
Base (b) = unknown
We apply the Pythagorean theorem to find the base (distance between the ladder and the bottom of the slide):
a^2 + b^2 = c^2
Substitute the known values into the equation:
2^2 + b^2 = 4^2
4 + b^2 = 16
Now solve for b^2:
b^2 = 16 - 4
b^2 = 12
Take the square root of both sides to solve for b:
b ≈ √12
b ≈ 3.46
Rounding to the nearest tenth, the distance between the ladder and the bottom of the slide is approximately 3.5 meters.
Alec likes to skate at an ice cream parlor that is due south of his school and due west of his favorite game store. If the ice cream parlor is 9 miles from his school and the straight-line distance between the school and the game store is 10 miles, how far is the ice cream parlor from the game store? If necessary, round to the nearest tenth.
The locations of the school, game store, and ice cream parlor form a right triangle. The ice cream parlor being due south of the school and due west of the game store means that the distances from the school to the ice cream parlor and from the ice cream parlor to the game store are the legs of the triangle, and the straight-line distance between the school and game store is the hypotenuse.
Using the Pythagorean theorem (a^2 + b^2 = c^2), where:
- a is the distance from the school to the ice cream parlor,
- b is the distance from the ice cream parlor to the game store (what we want to find),
- c is the straight-line distance from the school to the game store.
Given:
a = 9 miles
c = 10 miles
b = ?
We need to find b, so we rewrite the Pythagorean theorem as follows:
b^2 = c^2 - a^2
Now plug in the values:
b^2 = 10^2 - 9^2
b^2 = 100 - 81
b^2 = 19
Find the square root of b^2 to solve for b:
b ≈ √19
b ≈ 4.36 miles
Rounded to the nearest tenth, the ice cream parlor is approximately 4.4 miles from the game store.
This diagram shows a cube. Each edge of the cube is 13 units long. The diagonal of each face is x units long. The diagonal of the cube is y units long.
Find x and y. If necessary, round your answers to the nearest tenth.
To find the length of the diagonal of each face (x) of the cube, you can apply the Pythagorean theorem because the face of the cube is a square. The diagonal of the square divides it into two right-angled triangles, with the sides of the square being the legs, and the diagonal being the hypotenuse.
Given that each edge of the cube (side of the square) is 13 units, you can calculate x using the Pythagorean theorem like this:
a^2 + b^2 = x^2
Since a = b (because they are both sides of the square and equal in length), and they're both 13 units:
13^2 + 13^2 = x^2
169 + 169 = x^2
338 = x^2
Taking the square root of both sides to find x:
x = √338
x ≈ 18.384776
Rounded to the nearest tenth, x is 18.4 units.
Now, to find the length of the diagonal of the cube (y), you will again use the Pythagorean theorem. The diagonal of the cube creates a right-angled triangle with the side of the cube and the diagonal of the base (face) as the legs.
We already found x (the diagonal of each face), and we know that each side is 13 units, so:
x^2 + 13^2 = y^2
(18.4)^2 + 13^2 = y^2
338.56 + 169 = y^2
507.56 = y^2
Taking the square root of both sides to find y:
y = √507.56
y ≈ 22.526
Rounded to the nearest tenth, y is 22.5 units.
Find the distance between the points (
–
8,1) and (
–
2,
–
7).
If necessary, round your answer to the nearest tenth.
units
To find the distance between two points in a Cartesian coordinate system, you can use the distance formula which is derived from the Pythagorean theorem. The distance formula for two points (x1, y1) and (x2, y2) is:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Given the two points (-8, 1) and (-2, -7), you can plug these into the formula:
Distance = √[(-2 - (-8))^2 + (-7 - 1)^2]
Distance = √[(6)^2 + (-8)^2]
Distance = √[36 + 64]
Distance = √100
Distance = 10
Therefore, the distance between the points (-8, 1) and (-2, -7) is 10 units. There's no need to approximate because the result is an integer.
On the coordinate plane, rectangle WXYZ has vertices W(
–
3,
–
7), X(3,2), Y(6,0), and Z(0,
–
9).
What is the area of rectangle WXYZ? If necessary, round your answer to the nearest tenth.
The Island Loop Ferry takes the same route multiple times a day. It starts at the port of Seabreeze Village and travels due south for 8 miles to the town of Oceanview. From there, it travels due east for 15 miles to Portlandville. Finally, it makes a straight-line journey back to Seabreeze Village. If the ferry can travel 5 miles on a gallon of fuel, how many gallons does the ferry use to complete the route one time?
If necessary, round your answer to the nearest tenth.
gallons
Questions
Raggedy Rugs is holding their annual rug blowout sale this weekend! Since rugs will only be $6 per square foot, Katie is going to get a new rug for her apartment. The rug is shaped like a rectangle with a width of 8 feet and a diagonal length of 17 feet. How much will the rug cost?
$
To determine the cost of the rug, we first need to find its area in square feet. Since the rug is a rectangle with a known width and diagonal length, we can first work out the length of the rectangle by using the Pythagorean theorem, which relates the two sides of a right triangle to its hypotenuse (the diagonal of the rectangle, in this case).
The Pythagorean theorem:
a^2 + b^2 = c^2
Here, "a" and "b" are the sides of the triangle (rectangle's width and length), and "c" is the hypotenuse (diagonal of the rectangle):
Given:
Width (a) = 8 feet
Diagonal (c) = 17 feet
Length (b) = ?
We are looking to find the length "b":
b^2 = c^2 - a^2
b^2 = 17^2 - 8^2
b^2 = 289 - 64
b^2 = 225
Taking the square root of both sides to solve for b, we get:
b = √225
b = 15 feet
Now that we've found the length of the rectangle, we can calculate the area:
Area = Width × Length
Area = 8 feet × 15 feet
Area = 120 square feet
The cost per square foot for the rugs is $6. So, to calculate the cost of the rug:
Cost = Area × Price per square foot
Cost = 120 square feet × $6 per square foot
Cost = $720
Thus, Katie’s new rug will cost her $720.
This diagram shows right triangular prism MNOPQR.
What is the surface area of MNOPQR? If necessary, round your answer to the nearest tenth.
To calculate the surface area of a right triangular prism, we need to find the area of all its faces: the two triangular bases and the three rectangular sides.
However, without a diagram or additional information about the dimensions of triangular prism MNOPQR, we can't compute the exact surface area.
In the typical case of a right triangular prism, we would need to know:
1. The base and height (or lengths of all three sides) of the triangular bases.
2. The length of the prism (the altitude, which is the distance between the triangular bases).
The surface area would be calculated by adding the areas of the two triangular bases and the areas of the three rectangles that connect the corresponding sides of the two triangles.
If you can provide the specific dimensions of each side of the triangular base and the length of the prism, I could then calculate the surface area for you.
Find the distance between the points (
–
3,
–
3) and (6,9).
If necessary, round your answer to the nearest tenth.
To find the distance between two points on a coordinate plane, you can use the distance formula which is derived from the Pythagorean theorem:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Given the two points (-3, -3) and (6, 9), you can plug these into the formula:
Distance = √[(6 - (-3))^2 + (9 - (-3))^2]
Distance = √[(6 + 3)^2 + (9 + 3)^2]
Distance = √[9^2 + 12^2]
Distance = √[81 + 144]
Distance = √225
Distance = 15
Therefore, the distance between the points (-3, -3) and (6, 9) is 15 units. No rounding is necessary, as this is an exact answer.
△DEF is shown on the coordinate plane below.
What is the perimeter of △DEF? If necessary, round your answer to the nearest tenth.
units
To determine the perimeter of triangle DEF on the coordinate plane, you need to find the lengths of all three sides and then add them together to get the perimeter. However, since the coordinates of the points D, E, and F on the coordinate plane were not provided, I cannot compute the exact distances without that information.
If you have the coordinates of D, E, and F, you can find the distance between each pair of points using the distance formula:
Distance between two points (x1, y1) and (x2, y2) is given by:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Once you have the distances for each side:
1. DE (distance between points D and E),
2. EF (distance between points E and F),
3. FD (distance between points F and D),
You can find the perimeter by adding the lengths of all three sides:
Perimeter = DE + EF + FD
Please provide the coordinates for points D, E, and F, and I will calculate the perimeter for you.
d=(8,-8)
e=(5,9)
f= (-9,-5)
With the given coordinates for points D, E, and F, we can calculate the distances between each pair of points using the distance formula and then sum these distances to find the perimeter of triangle DEF.
The distance formula is:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Distance DE (between points D and E):
DE = √[(5 - 8)^2 + (9 - (-8))^2]
DE = √[(-3)^2 + (17)^2]
DE = √[9 + 289]
DE = √298
DE ≈ 17.3 units (rounded to the nearest tenth)
Distance EF (between points E and F):
EF = √[(-9 - 5)^2 + (-5 - 9)^2]
EF = √[(-14)^2 + (-14)^2]
EF = √[196 + 196]
EF = √392
EF ≈ 19.8 units (rounded to the nearest tenth)
Distance DF (between points D and F):
DF = √[(-9 - 8)^2 + (-5 - (-8))^2]
DF = √[(-17)^2 + (3)^2]
DF = √[289 + 9]
DF = √298
DF ≈ 17.3 units (rounded to the nearest tenth)
Now that we have the lengths of the sides, we can find the perimeter by adding them together:
Perimeter = DE + EF + DF
Perimeter ≈ 17.3 + 19.8 + 17.3
Perimeter ≈ 54.4 units
So the perimeter of triangle DEF is approximately 54.4 units, rounded to the nearest tenth.
A chessboard is one large square made up of 8 rows of 8 equal sized smaller squares. Krysta has two chessboards. On one board, the small squares are 2 inches by 2 inches. On the other, they are 3 inches by 3 inches. How much longer is the diagonal of the larger board than the diagonal of the smaller board?
To find the length of the diagonal of each chessboard, we first need to determine the total length of one side of the larger squares made by each board.
For the chessboard with small squares of 2 inches by 2 inches:
Since there are 8 small squares in a row, the total length of one side of the chessboard is:
8 squares * 2 inches/square = 16 inches.
For the chessboard with small squares of 3 inches by 3 inches:
Since there are 8 small squares in a row, the total length of one side of the chessboard is:
8 squares * 3 inches/square = 24 inches.
The diagonal (d) of a square can be found using the formula for the diagonal of a square, which is a special case of the Pythagorean theorem:
d = a * √2, where a is the side length of the square.
Diagonal of the smaller (2-inch) board:
d₁ = 16 inches * √2
Diagonal of the larger (3-inch) board:
d₂ = 24 inches * √2
Now, let's find the difference between the diagonals of the larger and smaller boards:
Difference = d₂ - d₁
Difference = (24 * √2) - (16 * √2)
Difference = 8 * √2 inches.
Now to find the numerical value:
Difference ≈ 8 * 1.414
Difference ≈ 11.312 inches.
Therefore, the diagonal of the larger board is approximately 11.3 inches longer than the diagonal of the smaller board when rounded to the nearest tenth.
Isosceles triangle JKL is shown below. Legs
JK
and
KL
are each 12 units long, and the height
KM
is 8 units.
What is the length of
JL
? If necessary, round your answer to the nearest tenth.
If ΔJKL is an isosceles triangle and KM is the height from K to JL, then KM bisects the base JL at point M. This means the base JL is divided into two equal lengths JM and ML.
We have a right-angled triangle JKM formed by the height KM and half of the base of the isosceles triangle as follows:
- JM (half of JL) is the base we want to find.
- JK is the hypotenuse and its length is 12 units.
- KM is the height and is 8 units long.
We use the Pythagorean theorem to solve for JM:
JK^2 = JM^2 + KM^2
Plugging in the known values:
12^2 = JM^2 + 8^2
144 = JM^2 + 64
JM^2 = 144 - 64
JM^2 = 80
JM = √80
JM ≈ 8.9 units (rounded to the nearest tenth)
Since JM is half of JL, to find the full length of the base JL, we double the length of JM:
JL = 2 * JM
JL = 2 * 8.944
JL ≈ 17.9 units (rounded to the nearest tenth)
Therefore, the length of JL is approximately 17.9 units.
This diagram shows rectangular prism LMNOPQRS and diagonal
LQ
.
What is the surface area of LMNOPQRS? If necessary, round your answer to the nearest tenth.
square centimeters
To calculate the surface area of a rectangular prism, we need to find the area of all its faces (two of each type due to opposite faces being equal) and then sum them up. The surface area (SA) is given by:
SA = 2lw + 2lh + 2wh
where:
l = length of the prism
w = width of the prism
h = height of the prism
However, if you've only mentioned a diagram showing rectangular prism LMNOPQRS and the diagonal LQ without providing the actual dimensions for the length, width, and height, I can't compute the surface area.
Please provide the specific dimensions for length, width, and height of the rectangular prism, and then I can do the calculation for you. Without the dimensions, it’s not possible to determine the surface area. If LQ is the space diagonal and its length is given, we still need at least two dimensions of the prism to determine the third one using the space diagonal formula, but that also isn't enough to find the surface area without knowing individual side lengths.
To find the area of rectangle WXYZ, we need to determine the lengths of its sides. If it is indeed a rectangle, then opposite sides are congruent (have the same length), and the area can be calculated by multiplying the length of one side by the length of the adjacent side.
Let's find the distances between W(–3, –7) and X(3, 2), which appear to be opposite corners of the rectangle, and the distances between X(3, 2) and Y(6, 0), which also seem to be opposite corners.
Using the distance formula for points W and X:
d(WX) = √[(3 - (–3))^2 + (2 - (–7))^2]
d(WX) = √[(3 + 3)^2 + (2 + 7)^2]
d(WX) = √[6^2 + 9^2]
d(WX) = √[36 + 81]
d(WX) = √117
Using the distance formula for points X and Y:
d(XY) = √[(6 - 3)^2 + (0 - 2)^2]
d(XY) = √[3^2 + (–2)^2]
d(XY) = √[9 + 4]
d(XY) = √13
However, these are not the lengths of the sides of the rectangle; these are the lengths of the diagonals since W and X, and X and Y are actually diagonal pairs due to their coordinates. Rectangles have congruent diagonals, so both WX and XY should be diagonal lengths. It seems that there might be a misunderstanding here; usually, to determine the area of a rectangle, you would need the lengths of adjacent sides (not diagonals).
If we check distances between possible adjacent corners instead:
W(–3, –7) and Z(0, –9) which are likely adjacent:
d(WZ) = √[(0- (–3))^2 + (-9 - (–7))^2]
d(WZ) = √[3^2 + (-2)^2]
d(WZ) = √[9 + 4]
d(WZ) = √13
X(3, 2) and Y(6, 0), which are also likely adjacent:
d(XY) as computed above is also √13. This establishes that √13 is indeed a side length.
Given that we now have two adjacent sides, we can find the area by squaring the length of one side since √13 is the length of a side and rectangles have congruent sides:
Area = side_length^2
Area = (√13)^2
Area = 13
So, the area of rectangle WXYZ is 13 square units. There is no need to round because 13 is a precise value.