We use the distance formula to find the length of each side. The distance formula between two points (x_1, y_1) and (x_2, y_2) is √((x_2 - x_1)² + (y_2 - y_1)²). We calculate the square of the distance for convenience.

We use the distance formula to find the length of each side. The distance formula between two points (x_1, y_1) and (x_2, y_2) is √((x_2 - x_1)² + (y_2 - y_1)²). We calculate the square of the distance for convenience.

Perform the calculations for CD². We see that CD² is also equal to AB² and BC².

Perform the arithmetic for AC². This is the squared length of the first diagonal.

After calculating BD², we observe that AC² and BD² are equal. This confirms that the diagonals are equal in length.

Show that the points $(1, 7)$ , $(4, 2)$ , $(-1, -1)$ and $(-4, 4)$ are the vertices of a square.

Answer: The points $(1, 7)$ , $(4, 2)$ , $(-1, -1)$ and $(-4, 4)$ are the vertices of a square because all four sides are equal in length and both diagonals are equal in length.

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 $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . 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 $BC^2$ is equal to $AB^2$ .

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 $CD^2$ . We see that $CD^2$ is also equal to $AB^2$ and $BC^2$ .

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 $DA^2$ , 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 $AC^2$ . 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 $BD^2$ , we observe that $AC^2$ and $BD^2$ 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.

More from this chapter

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Show that the points (1,7)(1, 7)(1,7), (4,2)(4, 2)(4,2), (−1,−1)(-1, -1)(−1,−1) and (−4,4)(-4, 4)(−4,4) are the vertices of a square.