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Magazine Chapter 10: Problem 10
The real numbers \(a, b, c,\) and \(d\) are positive. Consider the quadrilateral with vertices at \(R(0,0)\) \(S(a, 0), T(a, a),\) and \(V(0, a)\)Explain why \(R S T V\) is a square. CAN'T COPY THE GRAPH
Short Answer
Expert verified
\(RSTV\) is a square because it has equal sides and right angles.
Step by step solution
01
Identify the Coordinates
The quadrilateral has the following vertices: \(R(0,0)\), \(S(a,0)\), \(T(a,a)\), and \(V(0,a)\). All these points are in a Cartesian coordinate system.
02
Calculate the Length of Side RS
To find the length of side \(RS\), we use the distance formula between points \((x_1, y_1)\) and \((x_2, y_2)\): \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For \(R\) and \(S\), this becomes \(\sqrt{(a - 0)^2 + (0 - 0)^2} = a\).
03
Calculate the Length of Side ST
Using the distance formula between \(S(a, 0)\) and \(T(a, a)\), we have \(\sqrt{(a - a)^2 + (a - 0)^2} = a\).
04
Calculate the Length of Side TV
For \(T(a, a)\) and \(V(0, a)\), the distance is \(\sqrt{(0 - a)^2 + (a - a)^2} = a\).
05
Calculate the Length of Side VR
For the side \(VR\), from \(V(0, a)\) to \(R(0, 0)\), the distance is \(\sqrt{(0 - 0)^2 + (0 - a)^2} = a\).
06
Check for Equal Sides
Since \(RS = ST = TV = VR = a\), all sides of the quadrilateral are equal in length.
07
Verify the Right Angles
Each vertex connects vertical and horizontal lines, meaning each internal angle is \(90^\circ\). For instance, the vectors \(RS\) and \(ST\) are perpendicular because their slopes are 0 and undefined, respectively.
08
Conclusion
Since all sides are equal in length and all angles are right angles, \(RSTV\) is a square.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Quadrilaterals
A quadrilateral is a polygon with four sides and four vertices (corners). It can take various forms such as squares, rectangles, trapezoids, and rhombuses, each having unique properties. To identify a specific type of quadrilateral, we check the lengths of its sides and the measures of its angles.
- If all sides are equal and all angles measure 90 degrees, the quadrilateral is a square.
- If opposite sides are equal and all angles are 90 degrees, it is a rectangle.
- If only one pair of opposite sides are parallel, it's recognized as a trapezoid.
Cartesian Coordinate System
The Cartesian coordinate system is a method of plotting points in space using two perpendicular axes labeled usually as x (horizontal) and y (vertical). Each point is identified by an ordered pair \(x, y\).
- The x-coordinate represents the horizontal position while the y-coordinate shows the vertical position.
- This system helps in visually understanding spatial relationships between various points or forms like lines, shapes, and curves.
- For a shape plotted on this grid, you can quickly assess properties like length and area using mathematical formulas.
The Distance Formula
The distance formula is a crucial tool for finding the length between two points in a plane. It's derived from the Pythagorean theorem and is written as: \[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here's how it helps:
Here's how it helps:
- Calculates the exact distance by taking the difference in x and y values between two points.
- This value is a numerical measure of the straight line connecting the points, regardless of their position.
Identifying Right Angles
Right angles are one of the most significant criteria for defining squares and rectangles. A right angle measures exactly 90 degrees. To confirm right angles in a quadrilateral in a coordinate plane, examine the slopes of the connecting lines.
- Lines with slopes that are negative reciprocals of each other are perpendicular.
- For example, a horizontal line has a slope of 0, and a vertical line has an undefined slope; these two lines form a 90-degree angle at their intersection.
