Show that the points (1,7), (4,2), (−1,−1) and (−4,4) are the vertices of a square.
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Step-by-Step Solution
Step 1: Define the Vertices
Let's assign variable names to the given coordinates to make it easier to refer to them throughout the solution. We will label them as points A, B, C, and D in sequence.
Step 2: Calculate the Square of the Length of Side AB
We use the distance formula to find the length of each side. The distance formula between two points (x1,y1) and (x2,y2) is (x2−x1)2+(y2−y1)2. We calculate the square of the distance for convenience.
Step 3: Compute AB Squared
Substitute the x and y coordinates into the distance formula and perform the arithmetic operations. Squaring a negative number results in a positive number.
Step 4: Calculate the Square of the Length of Side BC
Next, we apply the distance formula to find the square of the length of the side BC using the coordinates of points B and C.
Step 5: Compute BC Squared
Substitute the coordinates of B and C and calculate the square of the distance. We observe that BC2 is equal to AB2.
Step 6: Calculate the Square of the Length of Side CD
Now, we find the square of the length of the side CD using the coordinates of points C and D. Be careful with the signs when subtracting negative numbers.
Step 7: Compute CD Squared
Perform the calculations for CD2. We see that CD2 is also equal to AB2 and BC2.
Step 8: Calculate the Square of the Length of Side DA
Finally, we calculate the square of the length of the side DA, connecting point D back to point A. This will complete the lengths of all four sides.
Step 9: Compute DA Squared
After computing DA2, we confirm that all four sides have equal squared lengths. This indicates that the quadrilateral is either a rhombus or a square.
Step 10: Calculate the Square of the Length of Diagonal AC
To distinguish between a rhombus and a square, we must check the lengths of the diagonals. A square has equal diagonals. Let's start with diagonal AC.
Step 11: Compute AC Squared
Perform the arithmetic for AC2. This is the squared length of the first diagonal.
Step 12: Calculate the Square of the Length of Diagonal BD
Now we calculate the square of the length of the second diagonal, BD, using the coordinates of points B and D.
Step 13: Compute BD Squared
After calculating BD2, we observe that AC2 and BD2 are equal. This confirms that the diagonals are equal in length.
Step 14: Verify Square Properties
Since all four sides are equal (AB=BC=CD=DA) and both diagonals are equal (AC=BD), the given points form a square. These are the defining properties of a square.