Geometry

We are asked to maximize the area of the rectangular portion of a playground enclosed by a running track with a total perimeter of 500 ft. Step 1: Identify Variables Let the length of the rectangular portion be and the width be , where is the radius of the semicircles. Since there are two semicircles at each end, their total circumference equals the circumference of a full circle, which is . Step 2: Perimeter Equation From the given total perimeter condition, Simplifying, Dividing the entire equation by 2, Step 3: Area of the Rectangular Portion The area of the rectangular portion is: From the perimeter equation: Now, Expanding: Step 4: Maximizing the Area To maximize the area, take the derivative and set it equal to zero: Set the derivative equal to zero: Since , Step 5: Finding Using , Step 6: Final Answer Correct Answer: (2) \ 125

Step 1: Find the point of intersection of the given lines.
The given lines are:
x - 2y + 3 = 0 ...(1)
2x - y - 1 = 0 ...(2)
Multiply equation (1) by 2:
2x - 4y + 6 = 0 ...(3)
Subtract equation (2) from equation (3):
(2x - 4y + 6) - (2x - y - 1) = 0
-3y + 7 = 0
y =
Substitute y = in equation (1):
x - 2( ) + 3 = 0
x - + = 0
x - = 0
x =
The point of intersection is ( , ).

Step 2: Let the equation of the line passing through the intersection point be.
The equation of the line passing through ( , ) is:
y - = m(x - )
3y - 7 = m(3x - 5)
3y - 7 = 3mx - 5m
3mx - 3y + 7 - 5m = 0 ...(4)

Step 3: Find the coordinates of A and B.
For point A (x-intercept), put y = 0 in equation (4):
3mx + 7 - 5m = 0
x =
So, A = ( , 0)
For point B (y-intercept), put x = 0 in equation (4):
-3y + 7 - 5m = 0
y =
So, B = (0, )

Step 4: Let the dividing point be (h, k).
Given that (h, k) divides AB in the ratio -2 : 3.
Using section formula:
h =
k =

Step 5: Eliminate m to find the locus.
From h = , we get hm = 5m - 7, so m(h - 5) = -7, and m = .
From k = , we get 3k = -14 + 10m, so 10m = 3k + 14, and m = .
Equating the two expressions for m:

70 = (5 - h)(3k + 14)
70 = 15k + 70 - 3hk - 14h
0 = 15k - 3hk - 14h
14h + 3hk - 15k = 0
Replace (h, k) with (x, y):
14x + 3xy - 15y = 0
Therefore, the equation of the locus is 14x + 3xy - 15y = 0.

Step 1: Understand the given expression.
We are given the expression: Here, and are the inradii, and is an angle of the triangle.

Step 2: Apply the standard result from triangle geometry.
From standard results in triangle geometry, we know: where is the circumradius of the triangle, and is the angle opposite side .

Conclusion:
Thus, the correct expression is:

Step 1: Understand the Given Condition Given , triangle is right-angled at .

Step 2: Recall Formulas for and For a right-angled triangle: where is the hypotenuse.

Step 3: Compute Substitute the values of and : However, since , is the hypotenuse, and . Thus:

Final Answer:

We are given the relation in triangle : Where: - is the semi-perimeter, - are the exradii corresponding to angles respectively.

Step 1: Expand and Simplify the Given Equation
By expanding both sides: Expanding each term:

Step 2: Identifying Key Relationships
Recall the exradius relations: From the given identity, we can derive the desired relation using known properties of triangles. The given identity implies a symmetrical relationship among the sides and their respective segments.

Step 3: Identifying the Required Relationship
By manipulating the relationship using trigonometric identities and known triangle properties,

Step 4: Conclusion
Thus,

Final Answer: (C)