The points A (9, 0), B (9, 6), C(-9, 6) and D(-9, 0) are the vertices of a

To determine the type of quadrilateral formed by the points A(9, 0), B(9, 6), C(-9, 6), and D(-9, 0), we will analyze the properties of the sides and angles using the distance formula. The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:

d = √((x₂ – x₁)² + (y₂ – y₁)²)

### Step 1: Calculate the lengths of all sides
1. Length of AB:
AB = √((9 – 9)² + (6 – 0)²)
= √(0² + 6²)
= √36
= 6

2. Length of BC:
BC = √((-9 – 9)² + (6 – 6)²)
= √((-18)² + 0²)
= √324
= 18

3. Length of CD:
CD = √((-9 – (-9))² + (0 – 6)²)
= √(0² + (-6)²)
= √36
= 6

4. Length of DA:
DA = √((9 – (-9))² + (0 – 0)²)
= √((9 + 9)² + 0²)
= √324
= 18

 Step 2: Analyze the side lengths
From the calculations:
– AB = CD = 6 (opposite sides are equal)
– BC = DA = 18 (opposite sides are equal)

Thus, the quadrilateral has opposite sides that are equal in length.

Step 3: Check if the angles are right angles
To confirm whether the angles are right angles, we calculate the slopes of adjacent sides and check if their product is -1 (indicating perpendicularity).

1. Slope of AB:
Slope of AB = (6 – 0) / (9 – 9) = undefined (vertical line)

2. Slope of BC:
Slope of BC = (6 – 6) / (-9 – 9) = 0 (horizontal line)

3. Slope of CD:
Slope of CD = (0 – 6) / (-9 – (-9)) = undefined (vertical line)

4. Slope of DA:
Slope of DA = (0 – 0) / (9 – (-9)) = 0 (horizontal line)

Since the slopes of adjacent sides (e.g., AB and BC, or BC and CD) indicate perpendicularity (one is vertical and the other is horizontal), all angles are right angles.

 Step 4: Conclusion
The quadrilateral has opposite sides equal and all angles are right angles. Therefore, it is a **rectangle**.

The correct answer is:
b) rectangle
This question related to Chapter 7 Mathematics Class 10th NCERT. From the Chapter 7 Coordinate Geometry. Give answer according to your understanding.

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