The figure shows two circles with centers Q and S. Both the circles share a common tangent at point R, and T is tangent to both circles at J and K, and LN is tangent to both circles at N and M. R If PR = RS = 12, what is KL? %3D A. 6 В. 9 C. V288 D. V360
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Question 31
![**Mathematics Problem:**
**Problem Statement:**
31. The figure shows two circles with centers \( Q \) and \( S \). Both circles share a common tangent at point \( R \), and \( \overrightarrow{LJ} \) is tangent to both circles at \( J \) and \( K \), and \( \overline{LN} \) is tangent to both circles at \( N \) and \( M \).
[Insert diagram of two circles with various labeled points here]
- The smaller circle has center \( Q \) and is tangent to the larger circle at point \( R \).
- The line segment \( \overline{PR} = \overline{RS} = 12 \).
- \( \overline{LJ} \) is a common tangent touching the smaller circle at \( J \) and the larger circle at \( K \).
- \( \overlap{LN} \) is another common tangent touching the smaller circle at \( M \) and the larger circle at \( N \).
- From point \( L \), the tangents \( LJ \) and \( LN \) extend outwards.
**Question:**
If \( PR = RS = 12 \), what is the length of \( KL \)?
**Choices:**
A. 6
B. 9
C. \( \sqrt{288} \)
D. \( \sqrt{360} \)
**Explanation of Diagram:**
In the diagram:
- Two circles are depicted, one smaller with center \( Q \) and one larger with center \( S \).
- Both circles touch at point \( R \).
- Two common external tangents are drawn: one through points \( J \) and \( K \) (denoted as \( \overline{LJ} \)), and another through points \( N \) and \( M \) (denoted as \( \overline{LN} \)).
- The common tangents (\( \overline{LJ} \) and \( \overline{LN} \)) intersect at a point labeled \( L \).
- The lengths \( PR \) and \( RS \) are given as 12.
**Solution:**
To solve for \( KL \), consider the geometric properties and distances involved due to the tangency conditions and the given lengths.
**Answer:** C. \( \sqrt{288} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc14456c8-c349-46c2-a794-2f6de4965e41%2Faad58bcc-666e-41a4-86c3-76eabe37aff0%2F9ullm9h_processed.png&w=3840&q=75)
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