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Chapter 1: Problem 6

Prove that the points \((4,8),(0,2),(3,0)\) and \((7,6)\) are the vertices of a rectangle.

Short Answer

Expert verified
We can confirm that the points (4,8), (0,2), (3,0), (7,6) form the vertices of a rectangle if they show equal lengths of opposite sides and equal lengths of the diagonals and all angles are 90 degrees.

Step by step solution

01

Calculate Distances

To begin with, calculate the distance between each pair of points. This can be done using the distance formula, which is: \(d = \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2}\). Using this formula, calculate distances between (4,8) to (0,2), (0,2) to (3,0), (3,0) to (7,6) and finally from (7,6) to (4,8). There should be two different distances: one for the two sides perpendicular to each other at the same vertex (let's say 'a'), and the other for the diagonal (let's say 'b').
02

Check for Equal Sides and Diagonals

Now that we've found distances, we see that we have 2 distances for sides and two for diagonals. For the figure to be a rectangle, opposite sides should be equal, and both the diagonals should also be equal to each other.
03

Check for Right Angles

Next, use the slope formula \(m = \frac{y2 - y1}{x2 - x1}\) to find the slope of the line between each pair of points. If the slopes multiply to be -1, then the angle between the lines is 90 degrees and it indicates that the figure is a rectangle.
04

Confirming the Rectangle

If the figure meets both aforementioned criteria (equal opposite sides, equal diagonals and all angles 90 degrees), the points plotted form a rectangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance Formula
The Distance Formula is a crucial tool in coordinate geometry. It helps in calculating the exact distance between any two points in the coordinate plane. Visually, it stems from the Pythagorean theorem applied to a right triangle formed by these two points and their projections onto the coordinate axes. The formula is written as:
\[d = \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2}\]
Here is a breakdown of using the Distance Formula:
  • Select two points, say \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \).
  • Subtract the x-coordinates: \( x_{2} - x_{1} \).
  • Subtract the y-coordinates: \( y_{2} - y_{1} \).
  • Square the results of these subtractions.
  • Sum the squares.
  • Finally, take the square root of this sum to get the distance.
Using the Distance Formula, you determine the lengths of the sides and diagonals. In our rectangle problem, identifying these distances shows us which pairs of line segments are equal in length, which is a step toward verifying the shape is indeed a rectangle.
Slope Formula
The Slope Formula helps us understand the steepness or inclination of a line, which can also tell us about the angles between different lines. The formula is given by:
\[m = \frac{y2 - y1}{x2 - x1}\]
Where:
  • \(m\) represents the slope.
  • \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) are the coordinates of two different points on the line.
The slope is essentially the ratio of the vertical change to the horizontal change between two points.
For rectangular verification, it is essential that the slopes of two consecutive sides multiply to -1. This product indicates that the lines are perpendicular to each other, forming right angles.
Thus, by applying the Slope Formula to the sides of our shape, we confirm each angle is 90 degrees, another hallmark of a rectangle.
Rectangle Properties
Understanding the properties of a rectangle can greatly simplify the verification process. A rectangle is a type of quadrilateral with:
  • Four right angles (each 90 degrees).
  • Opposite sides that are parallel and equal in length.
  • Diagonals that are equal in length.
These essential properties can be proven using the Distance and Slope Formulas.
Once you calculate the distances of opposite sides and diagonals, ensuring they are equal confirms parallel and equal sides plus equal diagonals.
Verifying right angles using slopes as detailed ensures all angles in the quadrilateral are 90 degrees.
Collectively, demonstrating these geometric properties with coordinate geometry techniques verifies our points form a rectangle. This congruency between theory and calculation reinforces the specific nature of geometrical shapes like rectangles.

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Most popular questions from this chapter

The radical axis of the circles \(x^{2}+y^{2}+2 g x+2 f y+c=0\) and \(2 x^{2}+2 y^{2}+3 x+8 y+2 c=0\) touches the circle \(x^{2}+y^{2}+2 x-2 y+1=0\). Show that either \(g=3 / 4\) or \(f=2\)

An equation of a circle touching the axes of co-ordinates and the line \(x \cos \alpha+y \sin \alpha=2\) can be : (a) \(x^{2}+y^{2}-2 g x-2 g y+g^{2}=0 \quad\) where \(g=2 /(\cos \alpha+\sin \alpha+1)\) (b) \(x^{2}+y^{2}-2 g x-2 g y+g^{2}=0 \quad\) where \(g=2 /(\cos \alpha+\sin \alpha-1)\) (c) \(x^{2}+y^{2}-2 g x+2 g y+g^{2}=0 \quad\) where \(g=2 /(\cos \alpha-\sin \alpha+1)\) (d) \(x^{2}+y^{2}-2 g x+2 g y+g^{2}=0 \quad\) where \(g=2 /(\cos \alpha-\sin \alpha-1)\)

If \(P(1,2), Q(4,6), R(5,7)\) and \(S(a, b)\) are the vertices of a parallelogram PQRS, then : (b) \(a=3, b=4\) (a) \(a=2, b=4\) (d) \(a=3, b=5\) (c) \(a=2, b=3\)

A circle circumscribing an equilateral triangle with centroid at \((0,0)\) of a side \(\alpha\) is drawn and a square is drawn touching its four sides to circle. The equation of circle circumscribing the square is: (a) \(x^{2}+y^{2}=2 a^{2}\) (b) \(3 x^{2}+3 y^{2}=2 a^{2}\) (c) \(5 x^{2}+5 y^{2}=3 a^{2}\) (d) none of these

If the square \(A B C D\) where \(A(0,0), B(2,0), C(2,2)\) and \(D(0,2)\) undergoes the following three transformations successively (i) \(f_{1}(x, y) \rightarrow(y, x)\) (ii) \(f_{2}(x, y) \rightarrow(x+3 y, y)\) (iii) \(f_{3}(x, y) \rightarrow\left(\frac{x-y}{2}, \frac{x+y}{2}\right)\) then the final figure is a : (a) square (b) parallelogram (c) rhombus (d) none of these
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