What is the most precise name for quadrilateral ABCD with vertices A(3, 1), B(4, 5), C(7, 5), and D(7, 2)?
A. kite
B. square
C. rhombus
D. parallelogram
The most precise name for quadrilateral ABCD is a kite. A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length adjacent sides that are not parallel to each other. In this case, AB = AD and BC = CD, but AB and CD are not parallel to each other, and neither are AD and BC.
To determine the most precise name for the given quadrilateral, we need to understand the definitions of the different types of quadrilaterals.
1. A kite is a quadrilateral with two pairs of adjacent sides that are congruent (equal in length).
2. A square is a quadrilateral with all four sides of equal length, and all four angles are right angles (90 degrees).
3. A rhombus is a quadrilateral with all four sides of equal length, but the angles are not necessarily right angles.
4. A parallelogram is a quadrilateral with opposite sides that are parallel.
To find the most precise name for quadrilateral ABCD, we can evaluate the given coordinates of its vertices.
Using the distance formula, we can calculate the lengths of its sides:
AB = sqrt((4 - 3)^2 + (5 - 1)^2) = sqrt(1^2 + 4^2) = sqrt(1 + 16) = sqrt(17)
BC = sqrt((7 - 4)^2 + (5 - 5)^2) = sqrt(3^2 + 0) = sqrt(9) = 3
CD = sqrt((7 - 7)^2 + (2 - 5)^2) = sqrt(0 + 9) = sqrt(9) = 3
DA = sqrt((3 - 7)^2 + (1 - 2)^2) = sqrt((-4)^2 + (-1)^2) = sqrt(16 + 1) = sqrt(17)
Based on the calculated side lengths, we have AB = DA = sqrt(17), and BC = CD = 3. Therefore, both pairs of opposite sides are not congruent, ruling out the options of kite and square.
Since all four sides are equal in length (AB = BC = CD = DA), the quadrilateral is a rhombus. Thus, the most precise name for quadrilateral ABCD is option C, a rhombus.
To determine the most precise name for quadrilateral ABCD, we can analyze its properties.
1. Kite: A kite is a quadrilateral with two pairs of adjacent congruent sides. However, quadrilateral ABCD has sides AB and AD of different lengths, so it cannot be a kite.
2. Square: A square is a quadrilateral with all four sides congruent and all four angles right angles. Since not all sides or angles of ABCD are congruent or right angles, it cannot be a square.
3. Rhombus: A rhombus is a quadrilateral with all four sides congruent. Quadrilateral ABCD does not have all sides congruent, so it cannot be a rhombus.
4. Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel. By examining the slopes of the lines connecting the vertices, we can determine if ABCD has opposite sides parallel.
- The slope of AB = (5 - 1) / (4 - 3) = 4
- The slope of CD = (2 - 5) / (7 - 7) = undefined
Since the slopes of AB and CD are not equal, the opposite sides are not parallel. Thus, ABCD cannot be a parallelogram.
Therefore, based on the given information, none of the options A (kite), B (square), C (rhombus), or D (parallelogram) are applicable.