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The circumcentre of a triangle is (3, 3). If its two vertices are (4, 6) and (0, 4) find the third vertex of the triangle.
- a)(2, 4)
- b)(6, 2)
- c)(2, 6)
- d)(4, 4)
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The circumcentre of a triangle is (3, 3). If its two vertices are (4, ...
Let P (3, 3) be the circumcentre of DABC, C(x, y) be the third vertex.
PA = PB = PC
PA2= PB2= PC2
PA2 = PC2
⇒ (4 - 3)2 + (6 - 3)2
= (x - 3)2 + (y - 3)2
⇒ 1 + 9 = x2 + 9 - 6x + y2 + 9 - 6y
⇒ x2 + y2 - 6x - 6y + 8 = 0 ...(1)
and PB2 = PC2
⇒ (0 - 3)2 + (4, -3)2 = (x - 3)2 + (y - 3)2
⇒ 9 + 1 = x2 + 9 - 6x + y2 - 6y + 9
Equations (1) and (2) are identical
(x - 3)2 + (y - 3 )2 = 10
So, (6, 2) is the required point.
PA = PB = PC
PA2= PB2= PC2
PA2 = PC2
⇒ (4 - 3)2 + (6 - 3)2
= (x - 3)2 + (y - 3)2
⇒ 1 + 9 = x2 + 9 - 6x + y2 + 9 - 6y
⇒ x2 + y2 - 6x - 6y + 8 = 0 ...(1)
and PB2 = PC2
⇒ (0 - 3)2 + (4, -3)2 = (x - 3)2 + (y - 3)2
⇒ 9 + 1 = x2 + 9 - 6x + y2 - 6y + 9
Equations (1) and (2) are identical
(x - 3)2 + (y - 3 )2 = 10
So, (6, 2) is the required point.
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The circumcentre of a triangle is (3, 3). If its two vertices are (4, ...
To find the third vertex of the triangle, we need to use the fact that the circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle.
Given information:
Circumcentre coordinates: (3, 3)
Vertex coordinates: (4, 6) and (0, 4)
Let's proceed step by step to find the third vertex.
Step 1: Find the midpoint of the line segment joining the two given vertices.
To find the midpoint, we use the midpoint formula:
Midpoint coordinates = ((x1 + x2)/2, (y1 + y2)/2)
Using the given coordinates, we have:
Midpoint coordinates = ((4 + 0)/2, (6 + 4)/2)
Midpoint coordinates = (2, 5)
Step 2: Find the slope of the line passing through the two given vertices.
To find the slope, we use the slope formula:
Slope = (y2 - y1)/(x2 - x1)
Using the given coordinates, we have:
Slope = (4 - 6)/(0 - 4)
Slope = -2/-4
Slope = 1/2
Step 3: Find the negative reciprocal of the slope obtained in Step 2.
The negative reciprocal of 1/2 is -2/1, which is -2.
Step 4: Find the equation of the perpendicular bisector passing through the midpoint.
We have the slope (-2) and the point (2, 5) through which the perpendicular bisector passes. Using the point-slope form of a line, we have:
y - y1 = m(x - x1)
y - 5 = -2(x - 2)
y - 5 = -2x + 4
y = -2x + 9
Step 5: Find the intersection point of the perpendicular bisector and the line passing through the circumcentre.
To find the intersection point, we need to solve the system of equations formed by the perpendicular bisector equation and the equation of the line joining the circumcentre and the third vertex.
The equation of the line joining the circumcentre (3, 3) and the third vertex (x, y) is:
(y - 3) = (y - 6)/(x - 4) * (x - 3)
Solving the system of equations:
Substituting the equation of the perpendicular bisector in the equation of the line joining the circumcentre and the third vertex, we have:
-2x + 9 - 3 = (y - 6)/(x - 4) * (x - 3)
-2x + 6 = (y - 6)/(x - 4) * (x - 3)
Substituting the coordinates of the circumcentre (3, 3) in the above equation, we have:
-2(3) + 6 = (3 - 6)/(3 - 4) * (3 - 3)
0 = (-3)/(-1) * 0
0 = 0
This indicates that the equation of the perpendicular bisector is the same as the equation of the line joining the circumcentre and the third vertex. Therefore, the third vertex lies on the perpendicular bisector.
Step 6: Substitute the x-coordinate of the
Given information:
Circumcentre coordinates: (3, 3)
Vertex coordinates: (4, 6) and (0, 4)
Let's proceed step by step to find the third vertex.
Step 1: Find the midpoint of the line segment joining the two given vertices.
To find the midpoint, we use the midpoint formula:
Midpoint coordinates = ((x1 + x2)/2, (y1 + y2)/2)
Using the given coordinates, we have:
Midpoint coordinates = ((4 + 0)/2, (6 + 4)/2)
Midpoint coordinates = (2, 5)
Step 2: Find the slope of the line passing through the two given vertices.
To find the slope, we use the slope formula:
Slope = (y2 - y1)/(x2 - x1)
Using the given coordinates, we have:
Slope = (4 - 6)/(0 - 4)
Slope = -2/-4
Slope = 1/2
Step 3: Find the negative reciprocal of the slope obtained in Step 2.
The negative reciprocal of 1/2 is -2/1, which is -2.
Step 4: Find the equation of the perpendicular bisector passing through the midpoint.
We have the slope (-2) and the point (2, 5) through which the perpendicular bisector passes. Using the point-slope form of a line, we have:
y - y1 = m(x - x1)
y - 5 = -2(x - 2)
y - 5 = -2x + 4
y = -2x + 9
Step 5: Find the intersection point of the perpendicular bisector and the line passing through the circumcentre.
To find the intersection point, we need to solve the system of equations formed by the perpendicular bisector equation and the equation of the line joining the circumcentre and the third vertex.
The equation of the line joining the circumcentre (3, 3) and the third vertex (x, y) is:
(y - 3) = (y - 6)/(x - 4) * (x - 3)
Solving the system of equations:
Substituting the equation of the perpendicular bisector in the equation of the line joining the circumcentre and the third vertex, we have:
-2x + 9 - 3 = (y - 6)/(x - 4) * (x - 3)
-2x + 6 = (y - 6)/(x - 4) * (x - 3)
Substituting the coordinates of the circumcentre (3, 3) in the above equation, we have:
-2(3) + 6 = (3 - 6)/(3 - 4) * (3 - 3)
0 = (-3)/(-1) * 0
0 = 0
This indicates that the equation of the perpendicular bisector is the same as the equation of the line joining the circumcentre and the third vertex. Therefore, the third vertex lies on the perpendicular bisector.
Step 6: Substitute the x-coordinate of the
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The circumcentre of a triangle is (3, 3). If its two vertices are (4, 6) and (0, 4) find the third vertex of the triangle.a)(2, 4)b)(6, 2)c)(2, 6)d)(4, 4)Correct answer is option 'B'. Can you explain this answer?
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