Find the circumcenter of triangleABC with A(1,6), B(1,4), and C (5,4)
A. 5,3
B. 3,5
C. 7,3
D. 1,7
To find the circumcenter of a triangle, we need to find the intersection of the perpendicular bisectors of the triangle's sides.
First, let's find the equation of the line that passes through the midpoint of AB and is perpendicular to AB. The midpoint of AB is ((1+1)/2, (6+4)/2) = (1, 5). The slope of AB is (4-6)/(1-1) = -2/0, which is undefined. Therefore, the line perpendicular to AB is vertical, and its equation is x = 1.
Next, let's find the equation of the line that passes through the midpoint of BC and is perpendicular to BC. The midpoint of BC is ((1+5)/2, (4+4)/2) = (3, 4). The slope of BC is (4-4)/(5-1) = 0/4 = 0. Therefore, the line perpendicular to BC is horizontal, and its equation is y = 4.
The point of intersection of the lines x = 1 and y = 4 is (1, 4), which is the circumcenter of triangle ABC.
Answer: D. 1,7
it has to be the middle and neither A B or C goes as far as 7
Apologies for the confusion in my previous response. Let's correct it:
To find the circumcenter of triangle ABC, we have to find the intersection point of the perpendicular bisectors of the triangle's sides.
First, let's find the midpoint of side AB. The coordinates of A are (1, 6) and the coordinates of B are (1, 4). The midpoint of AB is given by ((1+1)/2, (6+4)/2) = (1, 5).
Next, let's determine the slope of side AB. The coordinates of A and B are (1, 6) and (1, 4) respectively. The slope of AB is (4-6)/(1-1) = -2/0, which is undefined.
Since the slope of AB is undefined, the perpendicular bisector of AB is a vertical line passing through the midpoint (1, 5), and its equation is x = 1.
Now, let's find the midpoint of side BC. The coordinates of B are (1, 4) and the coordinates of C are (5, 4). The midpoint of BC is given by ((1+5)/2, (4+4)/2) = (3, 4).
Next, let's determine the slope of side BC. The coordinates of B and C are (1, 4) and (5, 4) respectively. The slope of BC is (4-4)/(5-1) = 0/4 = 0.
Since the slope of BC is 0, the perpendicular bisector of BC is a horizontal line passing through the midpoint (3, 4), and its equation is y = 4.
The intersection of the lines x = 1 and y = 4 is (1, 4), which is the circumcenter of triangle ABC.
Therefore, the correct answer is D. 1, 4.