All questions of Circles for SSC CGL Exam

Explanation:

Given:
- Two equal circles of radius 4 cm intersect each other such that each passes through the centre of the other.

Common Chord:
- When two circles intersect in such a way that each circle passes through the center of the other, the line joining the points of intersection is called a common chord.

Properties of Common Chord:
- The common chord is perpendicular to the line joining the centers of the circles.
- It bisects the line joining the centers of the circles.

Calculation:
- In this case, since the circles have the same radius and intersect at the center of each other, the common chord will be the diameter of each circle.
- The diameter of a circle is equal to twice the radius.
- Therefore, the length of the common chord is 2 * 4 cm = 8 cm.
Therefore, the length of the common chord is 8 cm.

Understanding the Problem
In this problem, we are given two intersecting chords AB and CD of a circle that meet externally at point P. We know the lengths of the chords and one segment created by the intersection.
Given Values:
- Length of chord AB = 6 cm
- Length of chord CD = 3 cm
- Length of segment PD = 5 cm
We need to find the length of segment PB.
Using the Power of a Point Theorem
According to the Power of a Point theorem, for two chords AB and CD intersecting at point P outside the circle:
(PA * PB) = (PC * PD)
Where:
- PA = PB (since AB = 6 cm, let PB = x, then PA = 6 - x)
- PC = PD (let PC = y, then PD = 5 cm)
Setting Up the Equation
1. Since AB = 6 cm:
- PA + PB = 6 cm
- PB = x
- PA = 6 - x
2. For chord CD:
- PD = 5 cm
- PC = 3 cm - PD = 3 cm - 5 cm = -2 cm (not possible, so we need to adjust our equation)
When we look carefully:
- Let PB = x
- Then PA = 6 - x
- PC = 3 - (5 cm) = -2 cm (incorrect)
Instead, we use:
(PA * PB) = (PC * PD)
Now, since we need to find PB, we can equate:
Final Calculation
Using the lengths given:
(6 - PB) * PB = (3 - PD) * PD
Substituting PD = 5 cm:
(6 - PB) * PB = (3 - 5) * 5
Solving gives us:
(6 - PB) * PB = -2 * 5 = -10
This will yield PB to be approximately 7.35 cm when calculated correctly.
Conclusion
Thus, the length of PB is approximately 7.35 cm, making option 'B' the correct answer.

Understanding the Problem
To find the length of the perpendicular drawn from the center of a circle to a side of an equilateral triangle inscribed in that circle, we need to consider a few geometric properties.
Radius of the Circle
- The radius of the circle (R) is given as 10 cm.
Properties of the Equilateral Triangle
- An equilateral triangle inscribed in a circle has its vertices on the circumference.
- The center of the circle is also the centroid of the triangle.
Height of the Equilateral Triangle
- The height (h) of an equilateral triangle can be calculated using the formula:
h = (sqrt(3)/2) * a, where 'a' is the length of a side of the triangle.
Length of the Side
- The length of the side (a) of the inscribed equilateral triangle is related to the radius (R) by:
a = R * sqrt(3).
- Substituting R = 10 cm, we get:
a = 10 * sqrt(3) cm.
Finding the Perpendicular Length
- The perpendicular from the center to a side of the triangle divides the triangle into two equal parts.
- The distance from the center to a side can be calculated as:
Perpendicular length = h/3 (since the centroid divides the height in a 2:1 ratio).
- Substituting h = (sqrt(3)/2) * a, we find:
Perpendicular length = ((sqrt(3)/2) * a) / 3 = (sqrt(3)/2) * (10 * sqrt(3)) / 3.
- This simplifies to 10 * (3/6) = 5 * sqrt(3) cm.
Conclusion
The length of the perpendicular drawn from the center to any side of the triangle is 5 * sqrt(3) cm.
However, since the question provides options and none matches this value, the correct answer is option 'D' (None of these).