The problem asks for the coordinates of the circumcenter of a right triangle defined by vertices P(0, 0), Q(8, 0), and R(0, 6).
Circumcenter Property of Right Triangles
A key property is that the circumcenter of a right triangle is always located at the midpoint of its hypotenuse.
Identifying the Hypotenuse
The vertices P(0, 0), Q(8, 0), and R(0, 6) form a right triangle with the right angle at P(0, 0), since PQ lies on the x-axis and PR lies on the y-axis, which are perpendicular.
- The side PQ connects (0, 0) and (8, 0).
- The side PR connects (0, 0) and (0, 6).
- The side QR connects (8, 0) and (0, 6). This is the hypotenuse.
Calculating the Circumcenter Coordinates
To find the circumcenter, we calculate the midpoint of the hypotenuse QR using the midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Using the coordinates of Q(8, 0) and R(0, 6): $x_M = \frac{8 + 0}{2} = \frac{8}{2} = 4$ $y_M = \frac{0 + 6}{2} = \frac{6}{2} = 3$
Therefore, the coordinates of the circumcenter are (4, 3).
Similar Questions
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Important Questions from Coordinate Geometry
What is the reflection of the point (-1, 5) in the line x = 1?
What are the co-ordinates of the centroid of a triangle, whose vertices are A(1, -5), B(-4, 0) and C(3, -4)?
Slope of the line AB is 4/3. Co-ordinates of points A and B are (x, -5) and (2, -3) respectively. What is the value of x?
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
