Three vertices of a kite are (1,8), (1,0), and (4,6). What is the fourth vertex?
make a sketch and reflect the point (4,6) in the vertical line x = 1 to get
(-2,6)
To find the fourth vertex of the kite, we can use the fact that a kite has two pairs of adjacent congruent sides.
Let's assume the fourth vertex of the kite is (x, y).
The given vertices are:
A(1,8), B(1,0), and C(4,6).
First, let's find the length of the two pairs of congruent sides.
The distance between points A(1,8) and B(1,0) can be found using the formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance AB = sqrt((1 - 1)^2 + (0 - 8)^2)
= sqrt(0 + 64)
= sqrt(64)
= 8
The distance between points A(1,8) and C(4,6) can also be found using the distance formula:
Distance AC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance AC = sqrt((4 - 1)^2 + (6 - 8)^2)
= sqrt(3^2 + (-2)^2)
= sqrt(9 + 4)
= sqrt(13)
Since a kite has two pairs of congruent adjacent sides, we can conclude that the distance between points A and B is equal to the distance between points A and C.
Therefore,
AB = AC
8 = sqrt(13)
Squaring both sides of the equation to remove the square root:
(8)^2 = (sqrt(13))^2
64 = 13
This is not true. Therefore, our assumption that A, B, C, and D form a kite is incorrect.
Hence, we cannot determine the fourth vertex of the kite with the given information.
To find the fourth vertex of the kite, we need to understand the properties of a kite. A kite is a quadrilateral with two pairs of congruent adjacent sides. This means that the fourth vertex will be on the same line as one of the sides of the kite.
First, let's identify the two pairs of congruent adjacent sides in the given kite. The given vertices are (1,8), (1,0), and (4,6).
Let's start by identifying the two sides that are of equal length. Looking at the given vertices, we can see that the segment between (1,8) and (1,0) is vertical, so the length of its sides is 8-0 = 8 units. Similarly, the segment between (1,8) and (4,6) is horizontal, so its sides' length is 4-1 = 3 units.
Now, we need to determine which side is on the same line as the fourth vertex. Since we know that one pair of sides is vertical and the other is horizontal, we can tell that the fourth vertex will have an x-coordinate or a y-coordinate that matches the corresponding pair of sides.
In this case, since the side between (1,8) and (1,0) is vertical, the fourth vertex must have an x-coordinate of 1 to be on the same line. Similarly, since the side between (1,8) and (4,6) is horizontal, the fourth vertex must have a y-coordinate of 6 to be on the same line.
Therefore, the fourth vertex of the kite is (1,6).
To summarize the steps to find the fourth vertex of a kite:
1. Identify the two pairs of congruent adjacent sides.
2. Determine which side is vertical and which side is horizontal.
3. Find the x-coordinate and y-coordinate that matches the corresponding pair of sides.
4. The fourth vertex will have these coordinates.