Using the vertices: B(-9,1), E(2,3), F(12, -2), G(1,-4). Determine whether BEFG is a rhombus, a

rectangle, or a square. List all that apply and show the steps

Have you considered plotting the points and take a look?

if not, you can always check the side lengths. If they are all equal, then you need to check for right angles, using the Pythagorean theorems. If a diagonal qualifies as an hypotenuse, then you have a square, not just a rhombus.

To determine whether the quadrilateral BEFG is a rhombus, a rectangle, or a square, we need to check the properties of each type of quadrilateral.

1. Rhombus:
A rhombus is a quadrilateral with all four sides equal in length.

2. Rectangle:
A rectangle is a quadrilateral with all angles equal to 90 degrees.

3. Square:
A square is a quadrilateral with all sides equal in length and all angles equal to 90 degrees.

To determine the length of each side, we can use the distance formula:

Distance formula:
For two points (x1, y1) and (x2, y2), the distance between them is given by:
d = √[(x2 - x1)^2 + (y2 - y1)^2]

Let's calculate the lengths of each side:

BE: d = √[(2 - (-9))^2 + (3 - 1)^2]
= √[(11)^2 + (2)^2]
= √[121 + 4]
= √125
= 5√5

EF: d = √[(12 - 2)^2 + (-2 - 3)^2]
= √[(10)^2 + (-5)^2]
= √[100 + 25]
= √125
= 5√5

FG: d = √[(1 - 12)^2 + (-4 - (-2))^2]
= √[(-11)^2 + (-2)^2]
= √[121 + 4]
= √125
= 5√5

GB: d = √[(-9 - 1)^2 + (1 - (-4))^2]
= √[(-10)^2 + (5)^2]
= √[100 + 25]
= √[125]
= 5√5

We can see that all four sides, BE, EF, FG, and GB, are of the same length, 5√5.

To determine whether the quadrilateral is a rhombus, rectangle, or square:
- If all sides are equal in length (5√5), then it is a rhombus.
- If all angles are equal to 90 degrees, then it is a rectangle.
- If it satisfies both conditions (equal sides and equal angles), then it is a square.

To determine the angles, we can use the slope formula:

Slope formula:
For two points (x1, y1) and (x2, y2), the slope between them is given by:
m = (y2 - y1) / (x2 - x1)

Let's calculate the slopes of consecutive sides:

Slope of line BE:
m = (3 - 1) / (2 - (-9))
= 2 / 11

Slope of line EF:
m = (-2 - 3) / (12 - 2)
= -5 / 10
= -1/2

Slope of line FG:
m = (-4 - (-2)) / (1 - 12)
= -2 / (-11)
= 2/11

Slope of line GB:
m = (1 - (-4)) / (-9 - 1)
= (5) / (-10)
= -1/2

The slopes of the consecutive sides are the same for BEFG, which means the angles between them are equal.

Based on the calculation of side lengths and angles, the quadrilateral BEFG is a square.

In summary, the quadrilateral BEFG is a square.

To determine whether the quadrilateral BEFG is a rhombus, rectangle, or square, we need to examine its properties.

Step 1: Find the lengths of all four sides.
- The distance formula can be used to find the length between two points:
- The length of BE = √[(x₂ - x₁)² + (y₂ - y₁)²]
- Substituting the values:
- BE = √[(2 - (-9))² + (3 - 1)²]
- BE = √[11² + 2²]
- BE = √[121 + 4]
- BE = √125
- BE ≈ 11.18
- Similarly, you can find the lengths of EF, FG, and GB.

Step 2: Determine if all four sides are equal.
- In a rhombus, all four sides are equal in length.
- In a rectangle, opposite sides are equal in length. This means BE = FG and GB = EF.
- In a square, all four sides are equal in length, just like a rhombus.

Step 3: Determine if opposite sides are parallel.
- To determine if opposite sides are parallel, we need to compare the slopes of opposite sides.
- The slope between two points can be found using the formula:
- slope = (y₂ - y₁) / (x₂ - x₁)
- Calculate the slopes of BE and FG, and GB and EF, and compare them.
- If the slopes are equal, then the opposite sides are parallel.

Step 4: Determine if any pair of consecutive sides are perpendicular.
- To determine if any pair of consecutive sides are perpendicular, we need to compare the slopes of consecutive sides.
- If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
- Calculate the slopes of BE and EF, FG and GB, and GB and BE, and compare them.

By following these steps, you can determine whether BEFG is a rhombus, rectangle, or square by examining the lengths of its sides, the parallelism of opposite sides, and the perpendicularity of consecutive sides.