Given three vertices (3,5), (0,0), and (6,0). Which of the following could be the fourth?
A (3,-5)
B (9,5)
C (-3,5)
D (-3,-5)
To determine which of the given options could be the fourth vertex, we can analyze the characteristics and properties of a quadrilateral.
A quadrilateral is a polygon with four sides. In order for the given points to form a valid quadrilateral, they must satisfy certain conditions. One of these conditions is that opposite sides of the quadrilateral must be parallel.
To check if opposite sides are parallel, we can use the slope formula. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Using the given three vertices:
Vertex 1: (3, 5)
Vertex 2: (0, 0)
Vertex 3: (6, 0)
Let's find the slopes of the two opposite sides:
Slope of side 1: Slope between Vertex 1 and Vertex 2
slope_1 = (0 - 5) / (0 - 3) = -5 / -3 = 5/3
Slope of side 2: Slope between Vertex 2 and Vertex 3
slope_2 = (0 - 0) / (6 - 0) = 0 / 6 = 0
Now, let's analyze each given fourth vertex option:
A) (3, -5)
To find the slope of side 3 connecting Vertex 1 and Option A:
slope_3 = (-5 - 5) / (3 - 3) = -10 / 0 = undefined
Since the slope of side 3 is undefined, Option A does not form a parallel side and cannot be the fourth vertex. Therefore, we can eliminate Option A.
B) (9, 5)
To find the slope of side 3 connecting Vertex 1 and Option B:
slope_3 = (5 - 5) / (9 - 3) = 0 / 6 = 0
The slope of side 3 is 0, which is equal to the slope of side 2. This confirms that side 3 is parallel to side 2. Therefore, Option B could be the fourth vertex.
C) (-3, 5)
To find the slope of side 3 connecting Vertex 1 and Option C:
slope_3 = (5 - 5) / (-3 - 3) = 0 / -6 = 0
The slope of side 3 is 0, which is equal to the slope of side 2. This confirms that side 3 is parallel to side 2. Therefore, Option C could be the fourth vertex.
D) (-3, -5)
To find the slope of side 3 connecting Vertex 1 and Option D:
slope_3 = (-5 - 5) / (-3 - 3) = -10 / -6 = 5/3
The slope of side 3 is 5/3, which is equal to the slope of side 1. This confirms that side 3 is parallel to side 1. Therefore, Option D could be the fourth vertex.
In conclusion, the possible fourth vertex options are:
B) (9, 5)
C) (-3, 5)
D) (-3, -5)
These options form a valid quadrilateral with the given three vertices.