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:
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Given points and , the slope of is:
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Since is a rectangle, lines and are perpendicular. The slope of will be the negative reciprocal of , which is .
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Using point and slope , we find the equation of line : The equation of is .
2. Coordinates of Point C:
- Point lies on line and . Substituting (equation of ) into : Substituting into : Therefore, .
3. Coordinates of Point B:
- Since and are opposite corners of the rectangle, and , , and are known, will share the -coordinate of and the -coordinate of , thus:
4. Area of Rectangle ABCD:
- The length of (or ) is the vertical distance between points and or and :
- The length of (or ) is the horizontal distance between points and :
- The area of the rectangle is:
Final Answers:
- Equation of CD:
- Coordinates of C:
- Coordinates of B:
- Area of ABCD: square units
Would you like further details or clarifications on any part? Here are some related questions to explore:
- How do you find the midpoint of a line segment?
- What is the formula for the distance between two points?
- How can you verify if two lines are perpendicular?
- What are the properties of a rectangle in the coordinate plane?
- 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
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