Question

\(A(-2,-1), B(1,0), C(4,3)\) and \(D(1,2)\) are the four points of a quadrilateral. The quadrilateral is a : (a) Square (b) Rhombus (c) Parallelogram (d) none of (a), (b),(c)

Short answer

(a) square (b) rhombus (c) parallelogram (d) none of (a), (b), (c) Upon analyzing the side lengths, opposite sides equalities, and angles, it was determined that the quadrilateral does not belong to any of the given options (square, rhombus, or parallelogram). Thus, the correct answer is: (d) none of (a), (b), (c)

Step-by-step solution

Calculate the distances between points

We will use the distance formula to find the lengths of the sides of the quadrilateral: For any two points \((x_1, y_1)\) and \((x_2, y_2)\), the distance formula is: $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$ Let's compute the lengths of the sides AB, BC, CD, and DA. Distance AB: $$d_{AB} = \sqrt{(1-(-2))^2 + (0-(-1))^2} = \sqrt{9 + 1} = \sqrt{10}$$ Distance BC: $$d_{BC} = \sqrt{(4-1)^2 + (3-0)^2} = \sqrt{9 + 9} = \sqrt{18}$$ Distance CD: $$d_{CD} = \sqrt{(1-4)^2 + (2-3)^2} = \sqrt{9 + 1} = \sqrt{10}$$ Distance DA: $$d_{DA} = \sqrt{((-2)-1)^2 + ((-1)-2)^2} = \sqrt{9 + 9} = \sqrt{18}$$

Check if opposite sides are equal

We have the lengths of the sides: $$d_{AB} = d_{CD} = \sqrt{10}$$ $$d_{BC} = d_{DA} = \sqrt{18}$$ Since the opposite sides are equal, the quadrilateral is a parallelogram.

Determine if the quadrilateral is a square, rhombus, or parallelogram

To further classify the quadrilateral, we need to check if the adjacent sides are equal (which would make it a rhombus) and if the angles are 90 degrees (which would make it a square). If neither condition is met, it remains a parallelogram. First, let's check if adjacent sides are equal: $$d_{AB} = \sqrt{10} \neq \sqrt{18} = d_{BC}$$ The adjacent sides are not equal, so it's not a rhombus or a square. Now we are left with only two options: either it's a parallelogram or none of the given options.

Calculate the angle between sides AB and BC

To further confirm if it's a parallelogram, we will calculate the angle between the sides AB and BC. If the angle is not 180 degrees, then the quadrilateral would not fall into any of the categories given in the options. We will use the scalar product (dot product) formula to find the angle. For any two vectors A and B, the scalar product formula is: $$A \cdot B = |A||B|\cos(\theta)$$ Vectors AB and BC can be represented as: $$\vec{AB} = \langle 1 + 2, 1 \rangle = \langle 3, 1 \rangle$$ $$\vec{BC} = \langle 4 - 1, 3 - 0 \rangle = \langle 3, 3 \rangle$$ Now, let's compute the dot product: $$\vec{AB} \cdot \vec{BC} = (3)(3) + (1)(3) = 9 + 3 = 12$$ We also need the magnitudes of the vectors: $$|\vec{AB}| = \sqrt{3^2 + 1^2} = \sqrt{10}$$ $$|\vec{BC}| = \sqrt{3^2 + 3^2} = \sqrt{18}$$ Now, we can find the cosine of the angle: $$\cos(\theta) = \frac{\vec{AB} \cdot \vec{BC}}{|\vec{AB}||\vec{BC}|} = \frac{12}{\sqrt{10}\sqrt{18}}$$ As the value of cosine is not 0, the angle between the two vectors is not 180 degrees. So, the quadrilateral doesn't satisfy the properties of a parallelogram.

Choosing the correct answer

Based on our analysis, the quadrilateral formed by the points \(A(-2,-1)\), \(B(1,0)\), \(C(4,3)\), and \(D(1,2)\) does not belong to any of the given options (square, rhombus, parallelogram). Therefore, the correct answer is: (d) none of (a), (b), (c)

Key concepts

Distance Formula

The distance formula is a key tool in geometry for finding the distance between two points in a plane. It's like a ruler for the coordinate plane and is derived from the Pythagorean theorem. This formula helps in calculating the length of a line segment defined by two points, with coordinates two points, whether they are labeled as \((x_1, y_1)\) and \((x_2, y_2)\). The formula itself is \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \].Let's break down this formula:

• \( x_2 - x_1 \): This represents the difference in the x-coordinates, the horizontal part of the distance.
• \( y_2 - y_1 \): This represents the difference in the y-coordinates, the vertical part of the distance.
• Squaring these differences ensures they are non-negative values, and it aligns with the Pythagorean theorem approach.
• Finally, you take the square root to fetch the actual distance.

For example, when applied to the points \(A(-2,-1)\) and \(B(1,0)\), it calculates the distance AB as \(\sqrt{10}\). This technique is fundamental in classifying shapes based on side lengths.

Parallelogram

A parallelogram is a simple yet fascinating geometric shape characterized by having opposite sides that are parallel and equal in length. It's one of the building blocks of geometry studies, as many other shapes stem from its basic structure. To identify a parallelogram, one can use the property that both pairs of opposite sides are equal. In the exercise, we see this with distances:

• \(d_{AB} = d_{CD} = \sqrt{10}\)
• \(d_{BC} = d_{DA} = \sqrt{18}\)

When both pairs of opposite sides are equal, it signals the possibility of a parallelogram. However, additional checks, such as angle verification, are necessary to fully classify the shape.A parallelogram can come in different forms like a rectangle, rhombus or square, depending on additional properties such as the angles or side lengths being equal across the board. This distinction makes learning about parallelograms pivotal in geometry.

Dot Product

Dot product is a powerful algebraic operation in geometry that helps determine angles between two vectors. This technique is widely used to study directional relations and is essential for finding out if angles meet certain criteria, such as being perpendicular or parallel. The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by the formula:\[A \cdot B = |A||B|\cos(\theta)\]where \( \theta \) is the angle between the vectors.To calculate it, we multiply corresponding components of the vectors and add them up. Like in the exercise:

• \(\vec{AB} = \langle 3, 1 \) and \(\vec{BC} = \langle 3, 3 \).
• The dot product \(\vec{AB} \cdot \vec{BC} = 3 \times 3 + 1 \times 3 = 12\).

By knowing the dot product and the magnitudes of the vectors, which are \(\sqrt{10}\) and \(\sqrt{18}\) respectively, we find out whether the vectors form a 90-degree angle or not. In this case, the cosine of angle shown was not zero, indicating a non-right angle.

Geometry Concepts

Geometry is a branch of mathematics centered around the properties and relations of points, lines, surfaces, and solids. It's all about shapes and spatial understanding. In a geometry setting, various types of quadrilaterals (four-sided polygons) are explored, each having unique properties. Common quadrilateral types include:

• Square: All sides equal, all angles right angles.
• Rectangle: Opposite sides equal, all angles right angles.
• Rhombus: All sides equal, opposite angles equal.
• Parallelogram: Opposite sides equal and parallel.
• Trapezoid: At least one pair of opposite sides parallel.

The classification of any shape hinges on these parameters. In practical applications, problems often involve verifying these definitions through calculations like side lengths or angles, employing techniques like the distance formula or dot product test. Classifying a quadrilateral can sometimes require the combination of multiple geometric principles like parallelism, congruence, and the properties of angles.