Problem 31
Question
Two sides of a rhombus are along the lines, \(x-y+1=0\) and \(7 x-y-5=0\). If its diagonals intersect at \((-1,-2)\), then which one of the following is a vertex of this rhombus? [2016] (a) \(\left(\frac{1}{3},-\frac{8}{3}\right)\) (b) \(\left(-\frac{10}{3},-\frac{7}{3}\right)\) (c) \((-3,-9)\) (d) \((-3,-8)\)
Step-by-Step Solution
Verified Answer
(b) \(\left(-\frac{10}{3},-\frac{7}{3}\right)\).
1Step 1: Find the equations of the diagonals
To find the equations of the diagonals, use the fact that each diagonal is perpendicular to one of the sides of the rhombus and passes through
((-1, -2)). Find the slopes of the given lines, which are 1 (for the line x-y+1=0) and 7 (for the line 7x-y-5=0). The perpendicular slopes are -1 and -1/7 respectively.
2Step 2: Find the equations of diagonals
Using the point-slope form, for
((m1=-1)):
(y+2) = -1(x+1), simplifies to y = -x - 1. For (m2 = -1/7): (y+2) = (-1/7)(x+1), simplifying gives: y = (-1/7)x - (15/7). Thus, the diagonal equations are y = -x - 1 and y = (-1/7)x - (15/7).
3Step 3: Solve for intersection points with diagonals
Set to find the intersection points of lines and diagonals. Solving y = -x - 1 with y - x + 1 = 0 gives intersection point as (0, -1); for y = -x - 1 and 7x - y - 5 = 0 is solved, point is (-3, -4). Similarly intersect y = (-1/7)x - (15/7) with the two lines: x-y+1=0 yields x = -4, y = -3; with 7x-y-5=0 yields x = 1/3, y = -8/3.
4Step 4: Match with options
Check which coordinates align with one of the given vertices. Note: the potential vertices plotted from intersection points are (0, -1), (-3, -4). Compare these derived points to those in optional answers and find closest match option.
Key Concepts
RhombusDiagonalsIntersection PointsSlopes of Lines
Rhombus
A rhombus is a special type of parallelogram where all sides are of equal length. This means that opposite sides are parallel, and opposite angles are equal. One interesting property of a rhombus is that its diagonals are not only lines of symmetry but also perpendicularly bisect each other.
A rhombus looks like a slanted square, and while it holds many properties of a parallelogram, the equality of its sides makes it unique. The diagonals divide the rhombus into four right-angle triangles, which can be very useful in geometric problems, including those involving coordinate geometry.
A rhombus looks like a slanted square, and while it holds many properties of a parallelogram, the equality of its sides makes it unique. The diagonals divide the rhombus into four right-angle triangles, which can be very useful in geometric problems, including those involving coordinate geometry.
Diagonals
Diagonals of a rhombus are quite special because:
For example, in the problem, the perpendicular slopes of the sides are calculated, and their negative inverses provide the slopes of the diagonals. Using the intersection point and the slope, you can derive the equations of the diagonals.
- They intersect perpendicularly.
- They bisect each other.
- They divide the rhombus into congruent triangles.
For example, in the problem, the perpendicular slopes of the sides are calculated, and their negative inverses provide the slopes of the diagonals. Using the intersection point and the slope, you can derive the equations of the diagonals.
Intersection Points
The intersection point of the diagonals of a rhombus holds significance. This point is the center of the rhombus, where both diagonals cross. It helps to determine the diagonal equations' validity, ensuring they bisect correctly at this central point.
To find the exact location of the intersection point in coordinate geometry problems, evaluate the midpoint of any two vertices when the diagonals aren't given. In this problem, the intersection is given at \((-1, -2)\). Using this, equations for diagonals can be verified or even constructed, as they must pass through this central point.
To find the exact location of the intersection point in coordinate geometry problems, evaluate the midpoint of any two vertices when the diagonals aren't given. In this problem, the intersection is given at \((-1, -2)\). Using this, equations for diagonals can be verified or even constructed, as they must pass through this central point.
Slopes of Lines
Slopes are fundamental in coordinate geometry, defining the angle of a line with respect to the horizontal. They provide vital information, especially in determining relationships like parallelism or perpendicularity.
A line's slope can be used to find perpendicular lines. If a line has a slope \( m \), then the slope of a perpendicular line is \(-1/m\). For rhombuses, this relation is often used to find the slopes of the diagonals given the slopes of the sides.
A line's slope can be used to find perpendicular lines. If a line has a slope \( m \), then the slope of a perpendicular line is \(-1/m\). For rhombuses, this relation is often used to find the slopes of the diagonals given the slopes of the sides.
- In this problem, the slopes given were 1 and 7 for the sides.
- Thus, the diagonals have slopes -1 and -1/7, found by taking the negative reciprocal.
Other exercises in this chapter
Problem 29
A straight line through a fixed point \((2,3)\) intersects the coordinate axes at distinct points \(P\) and \(Q\). If \(O\) is the origin and the rectangle OPRQ
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In a triangle \(A B C\), coordianates of \(A\) are \((1,2)\) and the equations of the medians through \(B\) and \(C\) are \(x+y=5\) and \(x=4\) respectively. Th
View solution Problem 32
A straight line through origin \(\mathrm{O}\) meets the lines \(3 \mathrm{y}=10-4 \mathrm{x}\) and \(8 x+6 y+5=0\) at points A and B respectively. Then \(O\) di
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If a variable line drawn through the intersection of the lines \(\frac{x}{3}+\frac{y}{4}=1\) and \(\frac{x}{4}+\frac{y}{3}=1\), meets the coordinate axes at \(\
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