Math Problem Statement
The points A(-3, -4) and C(5,4) are the ends of the diagonal of a rhombus ABCD. a Find the equation of the diagonal BD. b Given that the side BC has gradient 5/3, find the coordinates of B and hence of D.
Solution
To solve this problem, let's go step by step.
Part (a): Finding the Equation of the Diagonal
-
Find the midpoint of : Since and are the ends of a diagonal of the rhombus, the diagonals of a rhombus bisect each other at right angles. The midpoint of will be the point where both diagonals and intersect.
The midpoint of and is calculated as:
-
Calculate the slope of : The slope between and is:
-
Determine the slope of : Since is perpendicular to , the slope of , denoted , is the negative reciprocal of :
-
Write the equation of line : Using the point-slope form of a line with point and slope : Simplifying, we get: So, the equation of the diagonal is:
Part (b): Finding the Coordinates of Points and
-
Determine the slope of : We are given that the slope of is .
-
Set up the equations for point : Let the coordinates of be .
Since is the midpoint of , we can use the midpoint formula:
Solving these equations:
- For :
- For : So, the coordinates of are .
-
Find the coordinates of : Since is also the midpoint of , and the coordinates of are , we can apply the midpoint formula again to find (we’ll let the coordinates of be ):
Solving these:
- For :
- For :
So, the coordinates of are .
Summary of Results
-
The equation of the diagonal is:
-
The coordinates of are , and the coordinates of are .
Would you like a detailed explanation of any specific step, or do you have any questions?
Related Questions:
- How do you find the midpoint of a line segment in a coordinate plane?
- What is the significance of the diagonals in a rhombus and how do they relate to each other?
- How do you determine the equation of a line perpendicular to a given line?
- What is the process for finding the equation of a line given a slope and a point on the line?
- How can you verify if four points form a rhombus on a coordinate plane?
Tip: When solving for coordinates on a rhombus, always remember that the diagonals bisect each other at right angles and each diagonal divides the rhombus into two congruent triangles.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Line Equations
Midpoint Theorem
Perpendicular Lines
Slope
Rhombus Properties
Formulas
Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Slope Formula: m = (y2 - y1) / (x2 - x1)
Equation of a Line: y - y1 = m(x - x1)
Theorems
Diagonals of a Rhombus Bisect Each Other at Right Angles
Perpendicular Lines Have Negative Reciprocal Slopes
Suitable Grade Level
Grades 9-11
Related Recommendation
Finding the Equation of Diagonal BD in a Rhombus Given Points and Slope
Find Coordinates of B and D of Rhombus ABCD Given A and C
Calculate the Length of Diagonal AC in a Rhombus Using the Distance Formula
Find Coordinates of Midpoint and Vertices in Rhombus ABCD
Finding Midpoint and Coordinates of Square ABCD from Vertices A and C