Three vertices of a kite are (6, 8), (9, 11), and (12, 8). What are the coordinates of the fourth vertex?
one axis is the line y=8
(9,11) is 3 above the line, so the 4th point must be 3 below the line, at (9,5)
Oh, the fourth vertex is playing hide and seek with you! Maybe it's camera shy. But don't worry, I'll help you find it. So, let's see...the kite has two sides of equal length, right?
Well, if we take a look at the given coordinates, we notice that the first and third vertices share the same y-coordinate of 8. That means they should be opposite vertices of the kite. Therefore, the fourth vertex must also have a y-coordinate of 11 to maintain the equal length of the sides.
Now, we just need to find the x-coordinate. Since the kite has two pairs of sides that are equal in length, the second vertex (9, 11) must be equidistant from both the first vertex (6, 8) and the fourth vertex.
So, let's calculate the distance from (9, 11) to (6, 8) and from (9, 11) to the unknown fourth vertex. A little bit of math will lead us to the solution. Hang on a moment...
Drumroll, please! The coordinates of the fourth vertex are (6, 11). Ta-da!
To find the coordinates of the fourth vertex of the kite, we need to understand the properties of a kite. A kite is a quadrilateral with two pairs of adjacent congruent sides. In a kite, the longer diagonal is perpendicular to the shorter diagonal.
Let's label the given vertices:
Vertex A: (6, 8)
Vertex B: (9, 11)
Vertex C: (12, 8)
Now, let's find the midpoint of the line segment connecting vertices A and C.
Midpoint formula:
Midpoint_x = (x1 + x2) / 2
Midpoint_y = (y1 + y2) / 2
Midpoint_x = (6 + 12) / 2 = 18 / 2 = 9
Midpoint_y = (8 + 8) / 2 = 16 / 2 = 8
The midpoint of line segment AC is (9, 8).
Since the longer diagonal is perpendicular to the shorter diagonal, we can find the equation of the line passing through the midpoint of line segment AC and vertex B. This line will be perpendicular to the line AC.
Slope formula:
Slope = (y2 - y1) / (x2 - x1)
Slope_BC = (11 - 8) / (9 - 12) = 3 / (-3) = -1
To find the equation of the line passing through the midpoint of AC and vertex B, we can use the point-slope form of a line.
Point-Slope Form:
y - y1 = m(x - x1)
Substituting the values:
y - 8 = -1(x - 9)
Distributing:
y - 8 = -x + 9
Rearranging:
y = -x + 9 + 8
y = -x + 17
Now, we have the equation of the line passing through the midpoint of AC (9, 8) and vertex B (9, 11) as y = -x + 17.
To find the coordinates of the fourth vertex, we need to find the point where the line intersects with the line AC.
Substituting y = -x + 17 into the equation of line AC, we get:
-1 = -x + 17
Rearranging:
x = 17 + 1
x = 18
Substituting the value of x back into the equation of line AC, we get:
y = -18 + 17
y = -1
Therefore, the coordinate of the fourth vertex is (18, -1).
To determine the coordinates of the fourth vertex of the kite, we need to understand a few properties of a kite. In a kite, the sides that are adjacent to each other are congruent, and the diagonals intersect at a right angle.
In this case, we already have three vertices of the kite, which are (6, 8), (9, 11), and (12, 8). Let's call them A, B, and C respectively. Now, we can find the fourth vertex using the following steps:
1. First, find the midpoint of the line segment AB. To find the midpoint, average the x-coordinates and the y-coordinates of A and B.
Midpoint of AB = ((x-coordinate of A + x-coordinate of B) / 2, (y-coordinate of A + y-coordinate of B) / 2)
Midpoint of AB = ((6 + 9) / 2, (8 + 11) / 2)
= (15/2, 19/2)
= (7.5, 9.5)
2. Next, find the midpoint of the line segment BC using a similar method as before.
Midpoint of BC = ((x-coordinate of B + x-coordinate of C) / 2, (y-coordinate of B + y-coordinate of C) / 2)
Midpoint of BC = ((9 + 12) / 2, (11 + 8) / 2)
= (21/2, 19/2)
= (10.5, 9.5)
3. Now, we know that the diagonals of a kite intersect at a right angle. Therefore, the midpoint of the diagonals will be the same point.
Since the midpoint of AD is the same point as the midpoint of BC, we can equate them:
(7.5, 9.5) = (10.5, 9.5)
4. Finally, we can find the coordinates of the fourth vertex, D, by substituting the x and y coordinates we just obtained.
The coordinates of the fourth vertex are (x-coordinate, y-coordinate) = (x-coordinate of B + x-coordinate of C - x-coordinate of A, y-coordinate of B + y-coordinate of C - y-coordinate of A)
Coordinates of D = (9 + 12 - 6, 11 + 8 - 8)
= (15, 11)
Therefore, the coordinates of the fourth vertex are (15, 11).