Math Problem Statement

find the length of lines BC, CD, DE, and EB

Find the length of lines BC, CD, DE, and EB based on the coordinates provided in the graph.

Solution

To find the lengths of the lines BCBC, CDCD, DEDE, and EBEB, we need to use the distance formula between the points on the coordinate plane. The distance formula between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

From the image, we can estimate the coordinates of the points BB, CC, DD, and EE:

  • B(−3,5)B (-3, 5)
  • C(1,1)C (1, 1)
  • D(3,4)D (3, 4)
  • E(−1,7)E (-1, 7)

Now, let's apply the distance formula for each segment:

Length of BCBC:

= \sqrt{(1 + 3)^2 + (1 - 5)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.66$$ ### Length of $$CD$$: $$CD = \sqrt{(3 - 1)^2 + (4 - 1)^2} = \sqrt{(3 - 1)^2 + (4 - 1)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61$$ ### Length of $$DE$$: $$DE = \sqrt{(-1 - 3)^2 + (7 - 4)^2} = \sqrt{(-1 - 3)^2 + (7 - 4)^2} = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$ ### Length of $$EB$$: $$EB = \sqrt{(-3 - (-1))^2 + (5 - 7)^2} = \sqrt{(-3 + 1)^2 + (5 - 7)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$$ ### Summary of the lengths: - $$BC \approx 5.66$$ - $$CD \approx 3.61$$ - $$DE = 5$$ - $$EB \approx 2.83$$ Would you like further details or explanations on how the distance formula works? Here are some related questions you might find useful: 1. What is the distance formula, and how can it be derived from the Pythagorean theorem? 2. Can the coordinates of points B, C, D, and E be represented as vectors for a different approach? 3. How would the lengths change if the shape is rotated about the origin? 4. How can we verify if the figure is a parallelogram based on the side lengths? 5. How do the diagonals of the figure compare to the sides in terms of length? **Tip**: Always ensure that you accurately read coordinates from a graph to avoid small calculation errors in geometry problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Plane
Distance Formula

Formulas

Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10

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