Assertion (A) If A, B, C, D are four points on a semi-circular arc with centre at O such that |AB| =

Q: Why do we use position vectors from the center O?

Using the center $O$ as the origin simplifies the problem immensely because it allows us to use the geometric properties of the semi-circle, specifically the fact that $\overrightarrow{OD} = -\overrightarrow{OA}$.

Q: How did we get $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$?

This is the fundamental triangle law of vector addition. To go from $A$ to $B$, you can first go from $A$ to the origin $O$ (which is $-\overrightarrow{OA}$), and then from $O$ to $B$ (which is $\overrightarrow{OB}$). Adding them gives $\overrightarrow{OB} - \overrightarrow{OA}$.

Q: Why is $\overrightarrow{OD} = -\overrightarrow{OA}$?

Since $AD$ is the diameter of the semi-circle and $O$ is its center, the points $A$, $O$, and $D$ lie on a straight line. The distance from $O$ to $A$ is the same as $O$ to $D$ (both are radii), but they point in exactly opposite directions.

Q: What is the polygon law of vector addition?

The polygon law states that if you have a sequence of vectors connected head-to-tail (like $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$), their sum is the single vector drawn from the tail of the first vector to the head of the last vector (which is $\overrightarrow{AD}$).

Q: Why is Reason R incorrect if $2\overrightarrow{AD}$ is a valid mathematical step?

While $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{AD} = 2\overrightarrow{AD}$ is mathematically correct, Reason R claims this sum equals $2\overrightarrow{AO}$. Since $\overrightarrow{AD} = 2\overrightarrow{AO}$, the sum is actually $4\overrightarrow{AO}$. The mismatch in the coefficient makes the statement false.

Question

Assertion (A) If $A, B, C, D$ are four points on a semi-circular arc with centre at $O$ such that $|AB| = |BC| = |CD|$, then
$AB + AC + AD = 4AO + OB + OC$
Reason (R) Polygon law of vector addition yields
$AB + BC + CD + AD = 2AO$
In the light of the above statements, choose the most appropriate answer from the options given below.

Options:

(A) Both A and R are correct and R is the correct explanation of A.

(B) Both A and R are correct but R is not the correct explanation of A.

(C) A is correct but R is not correct.

(D) A is not correct but R is correct.

Sigma Percentile · Accepted Answer

Vector addition can sometimes feel like a maze of arrows pointing in every direction, but when you anchor your thoughts to a solid geometric foundation, the math flows beautifully. Let's embark on a thrilling journey to decode this problem involving a semi-circle and the elegant laws of vector addition.

### The Geometry of the Setup

Imagine a semi-circle where points $A, B, C,$ and $D$ lie gracefully on the arc, with $O$ acting as the steadfast center. The problem states that the distances $|AB|, |BC|,$ and $|CD|$ are all equal. While this symmetry is beautiful, the most critical piece of information lies in the endpoints: $A$ and $D$ form the diameter of this semi-circle.

Because $A, O,$ and $D$ lie on a straight line, the vector pointing from the center $O$ to $D$ is exactly the opposite of the vector pointing from $O$ to $A$. Mathematically, we write this as:

$$\overrightarrow{OD} = -\overrightarrow{OA}$$

This simple relationship is our **master key**. It will unlock the entire problem, so keep it firmly in mind!

### Deconstructing the Assertion

Let's put Assertion (A) to the test. We are asked to evaluate the sum of three vectors originating from $A$: $\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD}$.

To make sense of this, we need a common reference point. The center $O$ is the perfect candidate. By expressing each vector in terms of position vectors relative to $O$, we can break them down using the triangle law of vector addition:

$$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$$

$$\overrightarrow{AC} = \overrightarrow{OC} - \overrightarrow{OA}$$

$$\overrightarrow{AD} = \overrightarrow{OD} - \overrightarrow{OA}$$

### The Magic of Position Vectors

Now, let's add these three equations together. We group the positive terms on one side and collect the negative $\overrightarrow{OA}$ terms:

$$\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} = (\overrightarrow{OB} + \overrightarrow{OC} + \overrightarrow{OD}) - 3\overrightarrow{OA}$$

Here is where the magic happens! Remember our master key? We know that $\overrightarrow{OD} = -\overrightarrow{OA}$. Let's substitute this into our equation:

$$= \overrightarrow{OB} + \overrightarrow{OC} + (-\overrightarrow{OA}) - 3\overrightarrow{OA}$$

Combining the $\overrightarrow{OA}$ terms gives us:

$$= \overrightarrow{OB} + \overrightarrow{OC} - 4\overrightarrow{OA}$$

Since reversing the direction of a vector flips its sign, $-4\overrightarrow{OA}$ is exactly the same as $+4\overrightarrow{AO}$. Substituting this back in, we get:

$$= 4\overrightarrow{AO} + \overrightarrow{OB} + \overrightarrow{OC}$$

Look closely at this result! It perfectly matches the expression given in Assertion (A). Therefore, we can confidently conclude that **Assertion (A) is absolutely correct**.

### Testing the Reason with Polygon Law

Now, let's turn our attention to Reason (R). It invokes the **polygon law of vector addition**. This law states that if you travel along a path of vectors connected head-to-tail, the net displacement is simply the vector from your starting point to your final destination.

If we walk from $A$ to $B$, then $B$ to $C$, and finally $C$ to $D$, our net displacement is just the vector from $A$ to $D$. Mathematically:

$$\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} = \overrightarrow{AD}$$

Reason (R) asks us to evaluate the sum $\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CD} + \overrightarrow{AD}$. Using our polygon law result, we can substitute the first three terms:

$$\overrightarrow{AD} + \overrightarrow{AD} = 2\overrightarrow{AD}$$

### The Final Verdict

So, the left side of Reason (R) simplifies to $2\overrightarrow{AD}$. But Reason (R) claims this sum is equal to $2\overrightarrow{AO}$. Is this true?

Let's relate $\overrightarrow{AD}$ to $\overrightarrow{AO}$. Since $AD$ spans the entire diameter, its length is twice the radius. In vector terms, $\overrightarrow{AD}$ is composed of $\overrightarrow{AO}$ and $\overrightarrow{OD}$. Since $\overrightarrow{OD}$ points in the same direction as $\overrightarrow{AO}$ and has the same magnitude, $\overrightarrow{AD} = 2\overrightarrow{AO}$.

Therefore, substituting this into our sum:

$$2\overrightarrow{AD} = 2(2\overrightarrow{AO}) = 4\overrightarrow{AO}$$

The true sum is $4\overrightarrow{AO}$, but Reason (R) claims it is only $2\overrightarrow{AO}$. There is a clear mismatch! Because of this incorrect coefficient, **Reason (R) is false**.

In conclusion, Assertion (A) stands strong as a correct mathematical statement, while Reason (R) falls apart under scrutiny. The correct choice is that A is correct, but R is not correct.

Select Answer:

Visualized Solution

\text{Analyzing the Geometry}

\text{Evaluating Assertion (A)}

\text{Summing the Vectors}

\text{Applying the Diameter Constraint}

\text{Evaluating Reason (R)}

\text{Final Conclusion}

The Sigma Insight: Vector Addition, Subtraction, and Resolution

The Geometry of the Setup

Deconstructing the Assertion

The Magic of Position Vectors

Testing the Reason with Polygon Law

The Final Verdict

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