Answer :
Sure, let's go through the process of answering the given question step-by-step.
### Step 1: Plotting the Points
First, let's plot the points on a coordinate plane:
- [tex]\( A(1, 3) \)[/tex]
- [tex]\( B(3, 5) \)[/tex]
- [tex]\( C(9, 1) \)[/tex]
- [tex]\( D(7, -1) \)[/tex]
### Step 2: Connecting the Points
Next, connect the points A, B, C, and D in the following order to form a quadrilateral:
- Connect [tex]\( A \)[/tex] to [tex]\( B \)[/tex]
- Connect [tex]\( B \)[/tex] to [tex]\( C \)[/tex]
- Connect [tex]\( C \)[/tex] to [tex]\( D \)[/tex]
- Connect [tex]\( D \)[/tex] to [tex]\( A \)[/tex]
### Step 3: Identify the Type of Quadrilateral
To identify the type of quadrilateral, calculate the lengths of the sides and diagonals.
Given the coordinates of the points, we find:
- [tex]\( AB \approx 2.828 \)[/tex]
- [tex]\( BC \approx 7.211 \)[/tex]
- [tex]\( CD \approx 2.828 \)[/tex]
- [tex]\( DA \approx 7.211 \)[/tex]
Also, calculate the diagonals:
- [tex]\( AC \approx 8.246 \)[/tex]
- [tex]\( BD \approx 7.211 \)[/tex]
### Step 4: Analyzing the Quadrilateral
To determine the type of quadrilateral, let's look at the following criteria:
1. Side Lengths: If opposite sides are both equal and adjacent sides are both equal as well, it is likely a parallelogram, possibly a rectangle, square, or rhombus.
2. Diagonals:
- If diagonals are equal and bisect each other, it can be a rectangle or square.
- If diagonals bisect each other but are not equal, it's a parallelogram, and more specifically, if they are equal in length, it could be a rhombus.
- If diagonals do not bisect each other and aren't equal, it doesn't match any of the regular types mentioned above.
### Step 5: Conclusion
For the side lengths:
- [tex]\( AB \approx 2.828 \)[/tex]
- [tex]\( BC \approx 7.211 \)[/tex]
- [tex]\( CD \approx 2.828 \)[/tex]
- [tex]\( DA \approx 7.211 \)[/tex]
For the diagonals:
- [tex]\( AC \approx 8.246 \)[/tex]
- [tex]\( BD \approx 7.211 \)[/tex]
Because [tex]\( AB = CD \)[/tex] and [tex]\( BC = DA \)[/tex], and one pair of opposite sides are equal. We might consider it to be an isosceles trapezoid.
However, since the lengths of the diagonals are distinct ([tex]\( AC \not= BD \)[/tex]), we can further refine this classification.
To finalize the type of quadrilateral accurately, we look deeper into the characteristics:
1. It is not a rectangle, square, or rhombus because the diagonals aren't both equal and don't bisect each other.
2. It isn't an ordinary parallelogram because its opposite sides aren't equal nor do the diagonals bisect each other evenly.
Given the measurements and our steps, we identify this quadrilateral as an Isosceles Trapezoid, where only one pair of opposite sides is parallel (trapezoid rule) with one property of congruent non-parallel sides.
This detailed analysis supports our conclusion that the quadrilateral ABCD, defined by its vertices, is indeed an Isosceles Trapezoid.
### Step 1: Plotting the Points
First, let's plot the points on a coordinate plane:
- [tex]\( A(1, 3) \)[/tex]
- [tex]\( B(3, 5) \)[/tex]
- [tex]\( C(9, 1) \)[/tex]
- [tex]\( D(7, -1) \)[/tex]
### Step 2: Connecting the Points
Next, connect the points A, B, C, and D in the following order to form a quadrilateral:
- Connect [tex]\( A \)[/tex] to [tex]\( B \)[/tex]
- Connect [tex]\( B \)[/tex] to [tex]\( C \)[/tex]
- Connect [tex]\( C \)[/tex] to [tex]\( D \)[/tex]
- Connect [tex]\( D \)[/tex] to [tex]\( A \)[/tex]
### Step 3: Identify the Type of Quadrilateral
To identify the type of quadrilateral, calculate the lengths of the sides and diagonals.
Given the coordinates of the points, we find:
- [tex]\( AB \approx 2.828 \)[/tex]
- [tex]\( BC \approx 7.211 \)[/tex]
- [tex]\( CD \approx 2.828 \)[/tex]
- [tex]\( DA \approx 7.211 \)[/tex]
Also, calculate the diagonals:
- [tex]\( AC \approx 8.246 \)[/tex]
- [tex]\( BD \approx 7.211 \)[/tex]
### Step 4: Analyzing the Quadrilateral
To determine the type of quadrilateral, let's look at the following criteria:
1. Side Lengths: If opposite sides are both equal and adjacent sides are both equal as well, it is likely a parallelogram, possibly a rectangle, square, or rhombus.
2. Diagonals:
- If diagonals are equal and bisect each other, it can be a rectangle or square.
- If diagonals bisect each other but are not equal, it's a parallelogram, and more specifically, if they are equal in length, it could be a rhombus.
- If diagonals do not bisect each other and aren't equal, it doesn't match any of the regular types mentioned above.
### Step 5: Conclusion
For the side lengths:
- [tex]\( AB \approx 2.828 \)[/tex]
- [tex]\( BC \approx 7.211 \)[/tex]
- [tex]\( CD \approx 2.828 \)[/tex]
- [tex]\( DA \approx 7.211 \)[/tex]
For the diagonals:
- [tex]\( AC \approx 8.246 \)[/tex]
- [tex]\( BD \approx 7.211 \)[/tex]
Because [tex]\( AB = CD \)[/tex] and [tex]\( BC = DA \)[/tex], and one pair of opposite sides are equal. We might consider it to be an isosceles trapezoid.
However, since the lengths of the diagonals are distinct ([tex]\( AC \not= BD \)[/tex]), we can further refine this classification.
To finalize the type of quadrilateral accurately, we look deeper into the characteristics:
1. It is not a rectangle, square, or rhombus because the diagonals aren't both equal and don't bisect each other.
2. It isn't an ordinary parallelogram because its opposite sides aren't equal nor do the diagonals bisect each other evenly.
Given the measurements and our steps, we identify this quadrilateral as an Isosceles Trapezoid, where only one pair of opposite sides is parallel (trapezoid rule) with one property of congruent non-parallel sides.
This detailed analysis supports our conclusion that the quadrilateral ABCD, defined by its vertices, is indeed an Isosceles Trapezoid.
