(4, 5)'(9, 5) and (5,1) are three of a quadrilateral whose opposite sides are equal and parallel.there are three possibilities for the co-ordinates of the missing vertex.Find all three,and draw the three quadrilateral on a grid
one side is the horizontal line from (4,5) to (9,5). It is clearly 5 units long.
So, the other parallel side is also 5 units long, starting at (5,1)...
To find the missing vertex coordinates, we first need to understand the properties of the quadrilateral. Here, we are given that the opposite sides are equal and parallel. Let's analyze the given points and determine how we can find the missing vertex.
We know that the given coordinates are:
A: (4, 5)
B: (9, 5)
C: (5, 1)
Since the opposite sides are equal and parallel, it means that line AB is parallel to line CD, and line BC is parallel to line DA. Additionally, line AB is equal in length to line CD, and line BC is equal in length to line DA.
Now, let's consider the possible locations for the missing vertex, which we'll call D. Since line AB is parallel to line CD, the y-coordinate of D will be the same as the y-coordinate of C, which is 1. Thus, we have one possibility for D:
D: (x, 1)
Next, let's consider the parallel line BC. To find the x-coordinate of D, we can use the fact that line BC is parallel to line DA. To determine the length of BC, we can use the distance formula:
Length of BC = √[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the given points B and C:
Length of BC = √[(x - 9)^2 + (1 - 5)^2]
Since the length of AB is equal to the length of CD, the length of BC will also be equal to the length of DA. Therefore, we can calculate the length of DA using the distance formula with the given points A and D:
Length of DA = √[(4 - x)^2 + (5 - 1)^2]
We can set up an equation by equating the lengths of BC and DA:
√[(x - 9)^2 + (1 - 5)^2] = √[(4 - x)^2 + (5 - 1)^2]
Simplifying, squaring both sides of the equation:
(x - 9)^2 + 16 = (4 - x)^2 + 16
Expanding and simplifying:
x^2 - 18x + 81 + 16 = x^2 - 8x + 16 + 16
x^2 - 18x + 97 = x^2 - 8x + 32
10x = 65
x = 6.5
Therefore, we have found the second possibility for the missing vertex D:
D: (6.5, 1)
To find the third possibility, we use the fact that opposite sides are equal in length. If AB is equal to CD and BC is equal to DA, then AC (the diagonal) must also be equal to BD (the other diagonal).
Calculating the lengths of both diagonals using the given coordinates:
Length of AC = √[(9 - 4)^2 + (5 - 5)^2] = 5
Length of BD = √[(x - 4)^2 + (1 - 5)^2] = 5
Simplifying, squaring both sides of the equation:
(x - 4)^2 + 16 = 25
Expanding and simplifying:
x^2 - 8x + 16 + 16 = 25
x^2 - 8x + 7 = 0
Factorizing, or using the quadratic formula, we find:
(x - 7)(x - 1) = 0
This gives us two possibilities for the x-coordinate of D:
x = 7 or x = 1
Therefore, we have found the third and final possibilities for the missing vertex D:
D1: (7, 1)
D2: (1, 1)
To summarize, the three possible coordinates for the missing vertex D are:
D1: (6.5, 1)
D2: (7, 1)
D3: (1, 1)
To draw these quadrilaterals on a grid, plot the coordinates A(4, 5), B(9, 5), C(5, 1), and the three possible coordinates for D: D1(6.5, 1), D2(7, 1), D3(1, 1). Connect the corresponding points to form the quadrilaterals, and ensure that opposite sides are parallel and equal in length.