In A Society There Is A Circular Park Having Two Gates.the Gates Are Placedat Two PointsA (10,20)and B(50,50)as Shown In Figure Below Two Fountain Are Installed At Points P And Qon AB All Such That AP=PQ=QBfind Radius Of The Circle
Introduction
In a society, there is a circular park with two gates, placed at points A (10, 20) and B (50, 50). Two fountains are installed at points P and Q on AB, such that AP = PQ = QB. The problem requires finding the radius of the circle. This problem is a classic example of a geometric problem that can be solved using the concept of circles and their properties.
Understanding the Problem
To solve this problem, we need to understand the given information and the properties of circles. The two gates are placed at points A (10, 20) and B (50, 50), and the two fountains are installed at points P and Q on AB, such that AP = PQ = QB. This means that the distance from point A to point P is equal to the distance from point P to point Q, and the distance from point Q to point B is also equal.
Visualizing the Problem
To visualize the problem, we can draw a diagram showing the circular park, the two gates, and the two fountains. The diagram will help us understand the relationships between the different points and the circle.
Finding the Radius of the Circle
To find the radius of the circle, we need to use the concept of the circumcenter of a triangle. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect. In this case, the triangle is formed by the points A, B, and P.
Step 1: Find the Midpoint of AB
The first step is to find the midpoint of AB, which is the point where the perpendicular bisector of AB intersects. The midpoint of AB can be found using the formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
where (x1, y1) and (x2, y2) are the coordinates of points A and B.
Step 2: Find the Slope of AB
The next step is to find the slope of AB, which is the ratio of the vertical change to the horizontal change. The slope of AB can be found using the formula:
Slope = (y2 - y1) / (x2 - x1)
Step 3: Find the Equation of the Perpendicular Bisector
The next step is to find the equation of the perpendicular bisector of AB, which is the line that passes through the midpoint of AB and is perpendicular to AB. The equation of the perpendicular bisector can be found using the formula:
y - y1 = m(x - x1)
where m is the slope of AB, and (x1, y1) is the midpoint of AB.
Step 4: Find the Intersection Point of the Perpendicular Bisectors
The next step is to find the intersection point of the perpendicular bisectors of AB and AP. This point is the circumcenter of the triangle, and it is the center of the circle.
Step 5: Find the Radius of the Circle
The final step is to find the radius of the circle, which is the distance from the center of the circle to any point on the circle. The radius of the circle can be found using the formula:
Radius = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the center of the circle, and (x2, y2) is any point on the circle.
Solution
Now that we have the steps to find the radius of the circle, let's apply them to the given problem.
Step 1: Find the Midpoint of AB
The midpoint of AB is:
Midpoint = ((10 + 50)/2, (20 + 50)/2) = (30, 35)
Step 2: Find the Slope of AB
The slope of AB is:
Slope = (50 - 20) / (50 - 10) = 30 / 40 = 3/4
Step 3: Find the Equation of the Perpendicular Bisector
The equation of the perpendicular bisector of AB is:
y - 35 = -4/3(x - 30)
Step 4: Find the Intersection Point of the Perpendicular Bisectors
To find the intersection point of the perpendicular bisectors of AB and AP, we need to find the equation of the perpendicular bisector of AP. The midpoint of AP is:
Midpoint = ((10 + 30)/2, (20 + 35)/2) = (20, 27.5)
The slope of AP is:
Slope = (35 - 20) / (30 - 10) = 15 / 20 = 3/4
The equation of the perpendicular bisector of AP is:
y - 27.5 = 4/3(x - 20)
Now, we need to find the intersection point of the two perpendicular bisectors. We can do this by solving the system of equations:
y - 35 = -4/3(x - 30) y - 27.5 = 4/3(x - 20)
Solving the system of equations, we get:
x = 25 y = 32.5
Step 5: Find the Radius of the Circle
The radius of the circle is:
Radius = sqrt((25 - 30)^2 + (32.5 - 35)^2) = sqrt((-5)^2 + (-2.5)^2) = sqrt(25 + 6.25) = sqrt(31.25) = 5.59
Therefore, the radius of the circle is approximately 5.59 units.
Conclusion
In this article, we have discussed the problem of finding the radius of a circle in a society with a circular park and two gates. We have used the concept of the circumcenter of a triangle and the properties of circles to find the radius of the circle. The solution involves finding the midpoint of AB, the slope of AB, the equation of the perpendicular bisector of AB, the intersection point of the perpendicular bisectors, and the radius of the circle. The final answer is approximately 5.59 units.