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.