Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the circumcenter(1 point) Responses
(5,6)
(8,5)
(6,4)
(5,3)
The correct answer is (6,4).
To find the orthocenter of triangle XYZ, we need to find the intersection point of the altitudes of the triangle.
The altitude of a triangle is a perpendicular line from a vertex to the opposite side.
Let's find the equation of the altitude from vertex X to side YZ:
1. Find the slope of the line YZ:
slope_YZ = (y2 - y1) / (x2 - x1)
= (5 - 1) / (8 - 8)
= 4 / 0 (undefined)
The slope is undefined because the line YZ is vertical.
2. Since the altitude is perpendicular to YZ, its slope would be the negative reciprocal of the slope_YZ.
Therefore, the slope of the altitude from X is 0.
3. Since the point X(2, 5) lies on the altitude, we can write the equation of the altitude as:
y - y1 = m(x - x1)
y - 5 = 0(x - 2)
y - 5 = 0
y = 5
Therefore, the altitude from X is the horizontal line y = 5.
Now, let's find the equation of the altitude from vertex Y to side XZ:
1. Find the slope of the line XZ:
slope_XZ = (y2 - y1) / (x2 - x1)
= (5 - 1) / (2 - 8)
= 4 / -6
= -2/3
2. Since the altitude is perpendicular to XZ, its slope would be the negative reciprocal of the slope_XZ.
Therefore, the slope of the altitude from Y is 3/2.
3. Since the point Y(8, 5) lies on the altitude, we can write the equation of the altitude as:
y - y1 = m(x - x1)
y - 5 = (3/2)(x - 8)
y - 5 = (3/2)x - 12
y = (3/2)x - 7
Now, let's find the intersection point of the two altitudes:
Since the first altitude is a horizontal line y = 5, and the second altitude is a line with slope (3/2) and y-intercept -7, we can set the two equations equal to each other and solve for x:
5 = (3/2)x - 7
(3/2)x = 12
3x = 24
x = 8
Plugging this value back into either equation, we find that the y-coordinate is 5.
Therefore, the orthocenter of triangle XYZ is the point (8, 5).
The correct answer is (8, 5).
Triangle XYZ has vertices at X(2, 5), Y(8, 5) and Z(8, 1). Find the coordinates of the orthocenter of triangle XYZ.(1 point) Responses
(6,4)
(8,5)
(5,6)
(5,3)
To find the circumcenter of a triangle, you need to find the intersection of the perpendicular bisectors of the sides.
Let's go step by step to find the circumcenter of triangle XYZ with vertices X(2, 5), Y(8, 5), and Z(8, 1).
Step 1: Find the midpoints of the sides.
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
Midpoint(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)
a) Midpoint of XY:
Midpoint of XY = ((2 + 8) / 2, (5 + 5) / 2) = (5, 5)
b) Midpoint of YZ:
Midpoint of YZ = ((8 + 8) / 2, (5 + 1) / 2) = (8, 3)
c) Midpoint of ZX:
Midpoint of ZX = ((2 + 8) / 2, (5 + 1) / 2) = (5, 3)
Step 2: Find the slopes of the sides.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
Slope = (y2 - y1) / (x2 - x1)
a) Slope of XY:
Slope of XY = (5 - 5) / (8 - 2) = 0 / 6 = 0
b) Slope of YZ:
Slope of YZ = (1 - 5) / (8 - 8) = -4 / 0 (undefined since the denominator is 0)
c) Slope of ZX:
Slope of ZX = (1 - 5) / (2 - 8) = -4 / -6 = 2 / 3
Note: The slope of a perpendicular line to a given line is the negative reciprocal of its slope. So, the slope of a perpendicular line to XY is undefined, and the slope of a perpendicular line to ZX is -3/2.
Step 3: Find the equations of the perpendicular bisectors.
The equation of a line with slope m passing through the midpoint (x, y) is given by:
y - y1 = m(x - x1)
a) Perpendicular bisector of XY:
Since the slope of XY is 0 (horizontal line), the perpendicular bisector will be a vertical line passing through the midpoint (5, 5). Hence, the equation is x = 5.
b) Perpendicular bisector of ZX:
Since the slope of ZX is 2/3, the negative reciprocal is -3/2. The midpoint (5, 3) lies on this line. Hence, the equation is y - 3 = -3/2(x - 5).
Step 4: Find the coordinates of the circumcenter.
To find the circumcenter, we need to find the intersection point of the perpendicular bisectors.
Solving the equations x = 5 and y - 3 = -3/2(x - 5), we can find the coordinates of the intersection point.
x = 5
Substituting x = 5 into the second equation:
y - 3 = -3/2(5 - 5)
y - 3 = 0
y = 3
Therefore, the coordinates of the circumcenter are (5, 3).
Hence, the correct answer is (5, 3).