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Chapter 17: Problem 27

Quadrilateral ABCD has vertices A(16, 0), \(B(6,-5), C(-5,-7),\) and \(D(5,-2)\). What kind of quadrilateral is ABCD?

Short Answer

Expert verified
Quadrilateral ABCD is a rhombus.

Step by step solution

01

Calculate the Length of Each Side

To determine the type of quadrilateral, first calculate the distance between each pair of vertices using the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Calculate AB:\[AB = \sqrt{(6 - 16)^2 + (-5 - 0)^2} = \sqrt{100 + 25} = \sqrt{125}\]Calculate BC:\[BC = \sqrt{(-5 - 6)^2 + (-7 + 5)^2} = \sqrt{121 + 4} = \sqrt{125}\]Calculate CD:\[CD = \sqrt{(5 + 5)^2 + (-2 + 7)^2} = \sqrt{100 + 25} = \sqrt{125}\]Calculate DA:\[DA = \sqrt{(16 - 5)^2 + (0 + 2)^2} = \sqrt{121 + 4} = \sqrt{125}\] All sides are equal.
02

Calculate the Diagonals

Calculate the length of diagonals AC and BD to check for special properties:Calculate AC:\[AC = \sqrt{(-5 - 16)^2 + (-7 - 0)^2} = \sqrt{441 + 49} = \sqrt{490}\]Calculate BD:\[BD = \sqrt{(6 - 5)^2 + (-5 + 2)^2} = \sqrt{1 + 9} = \sqrt{10}\]Diagonal lengths are not equal.
03

Determine the Type of Quadrilateral

Since all four sides are equal and the diagonals are not equal, we determine that the quadrilateral is a rhombus.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula: Measuring the Gap
When we want to find the distance between two points on a plane, we use the Distance Formula. This is a crucial tool in coordinates geometry, especially when dealing with shapes.
  • It's represented as \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • This formula helps calculate the exact length between any two points, which are defined by their coordinates \((x_1, y_1)\) and \((x_2, y_2)\).
Understanding this formula allows us to determine the lengths of the sides and diagonals of a quadrilateral by applying it to each pair of points. In the exercise, by using the Distance Formula on vertices of a quadrilateral, we can measure each side accurately. This becomes essential in identifying the type of quadrilateral at hand.
Coordinates Geometry: Mapping Shapes
Coordinates Geometry is a branch of mathematics that fuses algebra and geometry, using a coordinate system to investigate geometric shapes.
  • It allows us to plot points on a graph in the form of \((x, y)\).
  • This plotting helps us visualize and calculate the properties of geometric figures and objects.
With coordinates geometry, finding the precise location and distance between points on a plane becomes feasible. In our exercise, coordinates \((16, 0)\), \((6,-5)\), \((-5,-7)\), and \((5,-2)\) are vital for calculating the distances between vertices, crucially aiding in shape identification.
Quadrilateral Properties: Understanding the Types
Quadrilaterals are polygons with four sides and are categorized based on side and angle measurements.
  • a) Rhombus: All sides are equal, but angles can vary.
  • b) Square: A special rhombus with all angles equal to 90 degrees.
  • c) Rectangle: Opposite sides equal, and all angles are 90 degrees.
  • d) Parallelogram: Opposite sides are equal and parallel.
In our scenario, calculating the distances of all four sides of a quadrilateral shows that they are equal. However, since the diagonals aren鈥檛 equal, it distinguishes the figure as a rhombus. By knowing each type's properties, you can easily identify them, making quadrilateral properties crucial in shape classification.
Diagonals: Key Indicators of Shape Type
Diagonals are the line segments that link non-adjacent vertices in a polygon, playing a significant role in characterizing quadrilaterals.
  • They provide insight into symmetry and side length relationships.
  • When diagonals are equal, it often implies certain characteristics like symmetry or specific angles.
For the exercise, after establishing that all sides of the quadrilateral are equal, calculating diagonals determines its specific type. Here, the UNEQUAL diagonals indicate the shape is a rhombus, not a square. A square would have equal diagonals, highlighting why examining diagonals is essential in distinguishing similar-looking quadrilaterals.

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Most popular questions from this chapter

Points \(A, B,\) and \(C\) have coordinates \(\mathrm{A}(-2,5), \mathrm{B}(1,3),\) and \(\mathrm{C}(7,-1)\). Do \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) determine a triangle?

Triangle ABC has vertices A(0, 0), B(5, 2), and \(C(7,-3)\). Show that \(\triangle \mathrm{ABC}\) is isosceles.

Quadrilateral ABCD has vertices A(16, 0), \(B(6,-5), C(-5,-7),\) and \(D(5,-2)\). Find the slope of each diagonal.

Triangle ABC has vertices A(6, 7), B(-4, 9), and \(\mathrm{C}(0,1)\). Find the slope of \(\overline{\mathrm{BC}}\).

Line segments are parallel iff they lie in parallel lines. Consider the points \(\mathrm{A}(3,1)\) \(\mathrm{B}(6,7), \mathrm{C}(0,-5),\) and \(\mathrm{D}(2,-1)\). Does \(\overline{\mathrm{AB}}\) appear to be parallel to \(\overline{\mathrm{CD}} ?\)
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