A square is a quadrilateral with four equal sides. The area of a square is equal to square of its sides. The diagonal of a square is a line segment joining any two non-consecutive side. Find the area of a square and the length of its diagonal whose vertices are (3,1), (-1,5), and (-1,-3). Draw the square.

Well, that square doesn't seem to be very cooperative, does it?

Anyway, let's tackle this mathematically! To find the area of a square, we just need to know the length of one side and then square it. So, let's figure out the length of the side first.

Using the distance formula, we can find the length of the sides. Let's take two consecutive vertices: (3,1) and (-1,5).

The length of the side is the square root of the sum of the squares of the differences in x and y coordinates. So, the length of the side (denoted as s) is:

s = √((3 - (-1))² + (1 - 5)²)
= √(4² + (-4)²)
= √(16 + 16)
= √32

Now that we know the length of the side, we can find the area by squaring it:

Area = s²
= (√32)²
= 32

So, the area of the square is 32 square units.

Now, let's move on to finding the length of the diagonal. We know that the diagonal of the square joins any two non-consecutive sides. In this case, we can take the diagonal that connects the vertices (3,1) and (-1,-3).

To find the length of the diagonal, we can again use the distance formula. Using these two points, the length of the diagonal (denoted as d) is:

d = √((3 - (-1))² + (1 - (-3))²)
= √(4² + 4²)
= √(16 + 16)
= √32

So, the length of the diagonal is √32 units.

Now, as for drawing the square, I'm afraid I can't help you there. As a bot, I don't possess artistic skills. Maybe you can give it a try and let your inner Picasso shine?

To find the area of the square formed by the given vertices, we need to determine the length of one side of the square.

Using the formula for distance between two points in a coordinate plane, we can find the length of a side of the square.

The distance between (3,1) and (-1,5) is given by:
d1 = sqrt((3 - (-1))^2 + (1 - 5)^2)
= sqrt(4^2 + (-4)^2)
= sqrt(16 + 16)
= sqrt(32)
= 4 * sqrt(2)

Thus, the length of one side of the square is 4 * sqrt(2).

To find the area of the square, we square the length of one side:
Area = (4 * sqrt(2))^2
= 16 * 2
= 32 square units.

Next, let's find the length of the diagonal of the square.

Since the diagonal of a square connects any two non-consecutive vertices, we can consider the distance between (3, 1) and (-1, -3).

The distance is given by:
d2 = sqrt((3 - (-1))^2 + (1 - (-3))^2)
= sqrt(4^2 + 4^2)
= sqrt(16 + 16)
= sqrt(32)
= 4 * sqrt(2)

Therefore, the length of the diagonal is 4 * sqrt(2).

To draw the square, we can use the given vertices and connect them with line segments.

Given vertices: (3,1), (-1,5), (-1,-3)

Drawing the square with sides parallel to the axes, we can connect these vertices in order:

(3,1) --- (-1,5)
| |
(-1,-3) --- (3,-3)

Thus, the square formed by the given vertices can be drawn as shown above.

To find the area of a square and the length of its diagonal, we can use the distance formula and solve some equations.

Step 1: Find the length of the sides of the square.
We can use the distance formula to find the length between two given points (x₁, y₁) and (x₂, y₂). In this case, we can find the length of the sides by calculating the distance between the given vertices.

Using the distance formula:
Length of side₁ = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((-1 - 3)² + (5 - 1)²)
= √((-4)² + (4)²)
= √(16 + 16)
= √32
≈ 5.66 (rounded to two decimal places)

Step 2: Find the area of the square.
The area of a square is equal to the square of its side length. Since we know the length of one side from step 1, we can use it to calculate the area.

Area of square = (side length)²
= (5.66)²
= 32 (rounded to two decimal places)

So, the area of the square is 32 square units.

Step 3: Find the length of the diagonal.
The diagonal of a square forms a right triangle with two sides equal to the length of a side. We can use the Pythagorean theorem to find the length of the diagonal.

Pythagorean theorem: c² = a² + b²

In this case, the length of a side is given by √32, and two sides of the right triangle are the length of the side.

Using the Pythagorean theorem:
Length of diagonal² = (side length)² + (side length)²
= (√32)² + (√32)²
= 32 + 32
= 64

Taking the square root of both sides:
Length of diagonal = √64
= 8

So, the length of the diagonal is 8.

Step 4: Draw the square.
We can plot the given vertices on a coordinate plane and connect them to form a square.

The given vertices are (3,1), (-1,5), and (-1,-3).
To draw the square, connect these vertices in order:
(3,1) to (-1,5)
(-1,5) to (-1,-3)
(-1,-3) to (3,1)

(3,1)
|
|
(-1, 5)----------|-----------(3, 1)
|
|
(-1, -3)

This is the square with vertices (3,1), (-1,5), (-1,-3), and (3,1).

So, the area of the square is 32 square units, and the length of its diagonal is 8 units.

(-1,5) and (-1,-3) are the ends of a vertical line. Since (3,1) is not at the same y-coordinate as either of those, the vertical line must be a diagonal. It has length=8.

Since the diagonal has length 8, its sides have length 4√2, giving an area of 32.

Knowing that the other diagonal is horizontal and has length=8, it should be easy to plot the missing vertex and draw the square.

For what it's worth, the diagonal is a line segment joining two non-consecutive vertices, not sides.