Verify that the shape with vertices at P(-5, 2), Q(-1, 3), R(-2, -1) and S(-6, -2) is a rhombus.
If a shape is a rhombus, then all its four sides are of an equal length.
The distance between two points is given by the distance formula:
√((x1 - x2)^2 + (y1 - y2)^2)
Where (x1, y1) and (x2, y2) are the two points.
Put in all possible pairs of the given sides into this formula to verify that the distances, and the lengths of the quadrilateral are the same.
To verify if the given shape is a rhombus, we need to check two conditions:
1. All sides of the shape are equal in length.
2. The diagonals of the shape bisect each other at right angles.
Let's begin by calculating the lengths of all four sides:
Side PQ:
Distance Formula: √((x2 - x1)^2 + (y2 - y1)^2)
P(-5, 2)
Q(-1, 3)
Distance PQ = √((-1 - (-5))^2 + (3 - 2)^2)
= √(4^2 + 1^2)
= √(16 + 1)
= √17
Side QR:
Q(-1, 3)
R(-2, -1)
Distance QR = √((-2 - (-1))^2 + (-1 - 3)^2)
= √((-1)^2 + (-4)^2)
= √(1 + 16)
= √17
Side RS:
R(-2, -1)
S(-6, -2)
Distance RS = √((-6 - (-2))^2 + (-2 - (-1))^2)
= √((-4)^2 + (-1)^2)
= √(16 + 1)
= √17
Side SP:
S(-6, -2)
P(-5, 2)
Distance SP = √((-5 - (-6))^2 + (2 - (-2))^2)
= √((1)^2 + (4)^2)
= √(1 + 16)
= √17
Now we need to check if the diagonals bisect each other at right angles. For this, we can calculate the slopes of the diagonals:
Diagonal PR:
P(-5, 2)
R(-2, -1)
Slope PR = (y2 - y1)/(x2 - x1)
= (-1 - 2)/(-2 - (-5))
= (-3)/(-2 + 5)
= (-3)/3
= -1
Diagonal QS:
Q(-1, 3)
S(-6, -2)
Slope QS = (y2 - y1)/(x2 - x1)
= (-2 - 3)/(-6 - (-1))
= (-5)/(-6 + 1)
= (-5)/(-5)
= 1
Since the slopes of the diagonals are negative reciprocals of each other (the product of two perpendicular lines), we can conclude that the diagonals intersect at right angles.
Finally, we have determined that the shape with vertices P(-5, 2), Q(-1, 3), R(-2, -1), and S(-6, -2) satisfies both conditions of a rhombus. Therefore, the shape is a rhombus.