Math Problem Statement

The diagram shows a rectangle ABCD. the coordinate of A and D are (4,1) and (16,5) and the equation of AC is Y= x-3 find the equation of CD, coordinate of C, Coordinate of B and Area of ABCD

Solution

Let's solve this step-by-step:

1. Equation of Line CD:

  • Given points A(4,1)A(4, 1) and D(16,5)D(16, 5), the slope of ADAD is: mAD=51164=412=13m_{AD} = \frac{5 - 1}{16 - 4} = \frac{4}{12} = \frac{1}{3}

  • Since ABCDABCD is a rectangle, lines ADAD and CDCD are perpendicular. The slope of CDCD will be the negative reciprocal of 13\frac{1}{3}, which is 3-3.

  • Using point D(16,5)D(16, 5) and slope 3-3, we find the equation of line CDCD: y5=3(x16)y - 5 = -3(x - 16) y5=3x+48y - 5 = -3x + 48 y=3x+53y = -3x + 53 The equation of CDCD is y=3x+53y = -3x + 53.

2. Coordinates of Point C:

  • Point CC lies on line CDCD and ACAC. Substituting y=x3y = x - 3 (equation of ACAC) into y=3x+53y = -3x + 53: x3=3x+53x - 3 = -3x + 53 x+3x=56x + 3x = 56 4x=56    x=144x = 56 \implies x = 14 Substituting x=14x = 14 into y=x3y = x - 3: y=143=11y = 14 - 3 = 11 Therefore, C(14,11)C(14, 11).

3. Coordinates of Point B:

  • Since BB and CC are opposite corners of the rectangle, and A(4,1)A(4, 1), D(16,5)D(16, 5), and C(14,11)C(14, 11) are known, BB will share the xx-coordinate of AA and the yy-coordinate of CC, thus: B(4,11)B(4, 11)

4. Area of Rectangle ABCD:

  • The length of ABAB (or CDCD) is the vertical distance between points BB and AA or DD and CC: AB=111=10AB = 11 - 1 = 10
  • The length of ADAD (or BCBC) is the horizontal distance between points AA and DD: AD=164=12AD = 16 - 4 = 12
  • The area of the rectangle is: Area=AB×AD=10×12=120\text{Area} = AB \times AD = 10 \times 12 = 120

Final Answers:

  1. Equation of CD: y=3x+53y = -3x + 53
  2. Coordinates of C: (14,11)(14, 11)
  3. Coordinates of B: (4,11)(4, 11)
  4. Area of ABCD: 120120 square units

Would you like further details or clarifications on any part? Here are some related questions to explore:

  1. How do you find the midpoint of a line segment?
  2. What is the formula for the distance between two points?
  3. How can you verify if two lines are perpendicular?
  4. What are the properties of a rectangle in the coordinate plane?
  5. How do you calculate the slope of a line from its equation?

Tip: Remember, perpendicular lines have slopes that are negative reciprocals of each other!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Equations of Lines
Properties of Rectangles
Area Calculation

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y - y1 = m(x - x1)
Area of a rectangle: Area = length × width

Theorems

Perpendicular lines have slopes that are negative reciprocals

Suitable Grade Level

Grades 9-10