Math Problem Statement

The points B(-5, 0), C(-1, -3), D(2, 1), and E(-2, 4) form a quadrilateral. Find the desired slopes and lengths, then fill in the words that BEST identify the type of quadrilateral.

Solution

Let's solve this step-by-step:

Points provided:

  • B(5,0)B(-5, 0)
  • C(1,3)C(-1, -3)
  • D(2,1)D(2, 1)
  • E(2,4)E(-2, 4)

We need to calculate:

  1. Slopes of the lines: BC,CD,DE,EB\overline{BC}, \overline{CD}, \overline{DE}, \overline{EB}
  2. Lengths of the line segments: BC,CD,DE,EB\overline{BC}, \overline{CD}, \overline{DE}, \overline{EB}

Step 1: Formula Recap

  1. Slope formula:
    m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}
  2. Distance formula:
    d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Step 2: Compute each slope and length

Slope and length of BC\overline{BC}:

  • Points: B(5,0),C(1,3)B(-5, 0), C(-1, -3)
  • Slope: mBC=301(5)=34m_{BC} = \frac{-3 - 0}{-1 - (-5)} = \frac{-3}{4}
  • Length: dBC=(1(5))2+(30)2=(4)2+(3)2=16+9=25=5d_{BC} = \sqrt{(-1 - (-5))^2 + (-3 - 0)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Slope and length of CD\overline{CD}:

  • Points: C(1,3),D(2,1)C(-1, -3), D(2, 1)
  • Slope: mCD=1(3)2(1)=43m_{CD} = \frac{1 - (-3)}{2 - (-1)} = \frac{4}{3}
  • Length: dCD=(2(1))2+(1(3))2=(3)2+(4)2=9+16=25=5d_{CD} = \sqrt{(2 - (-1))^2 + (1 - (-3))^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Slope and length of DE\overline{DE}:

  • Points: D(2,1),E(2,4)D(2, 1), E(-2, 4)
  • Slope: mDE=4122=34=34m_{DE} = \frac{4 - 1}{-2 - 2} = \frac{3}{-4} = -\frac{3}{4}
  • Length: dDE=(22)2+(41)2=(4)2+(3)2=16+9=25=5d_{DE} = \sqrt{(-2 - 2)^2 + (4 - 1)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Slope and length of EB\overline{EB}:

  • Points: E(2,4),B(5,0)E(-2, 4), B(-5, 0)
  • Slope: mEB=045(2)=43=43m_{EB} = \frac{0 - 4}{-5 - (-2)} = \frac{-4}{-3} = \frac{4}{3}
  • Length: dEB=(5(2))2+(04)2=(3)2+(4)2=9+16=25=5d_{EB} = \sqrt{(-5 - (-2))^2 + (0 - 4)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Step 3: Identify the quadrilateral

  • Slopes of BC=34\overline{BC} = -\frac{3}{4}, CD=43\overline{CD} = \frac{4}{3}, DE=34\overline{DE} = -\frac{3}{4}, EB=43\overline{EB} = \frac{4}{3}
  • Lengths of BC=5\overline{BC} = 5, CD=5\overline{CD} = 5, DE=5\overline{DE} = 5, EB=5\overline{EB} = 5

This suggests that the quadrilateral is a rhombus, as:

  1. All sides are equal.
  2. Opposite slopes are equal.

Final Answer:

  • Slope of BC\overline{BC}: 34-\frac{3}{4}, Length of BC\overline{BC}: 55
  • Slope of CD\overline{CD}: 43\frac{4}{3}, Length of CD\overline{CD}: 55
  • Slope of DE\overline{DE}: 34-\frac{3}{4}, Length of DE\overline{DE}: 55
  • Slope of EB\overline{EB}: 43\frac{4}{3}, Length of EB\overline{EB}: 55
  • Quadrilateral BCDEBCDE is a rhombus.

Would you like to see the calculations for better understanding or have additional questions?


Related Questions:

  1. How do you determine if a quadrilateral is a rectangle using slopes?
  2. What are the characteristics of a rhombus in terms of diagonals?
  3. How do you calculate the area of a rhombus given its diagonals?
  4. What makes a square different from a rhombus?
  5. Can a parallelogram be a rhombus? Explain why.

Tip:

When identifying quadrilaterals, always compare both side lengths and slopes for a thorough analysis.

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Math Problem Analysis

Mathematical Concepts

Geometry
Analytic Geometry
Properties of Quadrilaterals

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Properties of rhombuses: equal side lengths, opposite sides parallel

Suitable Grade Level

Grades 9-12