Question

In a \(\Delta ABC\). with usual notation, match the items in List - I with the items in List - II and choose the correct option.

List I		List II
(A)	\(r_1r_2\sqrt{\bigg(\frac{4R-r_1-r_2}{r_1+r_2}\bigg)}\)	1	\(b\)
(B)	\(\frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}\)	2	\(a^2,b^2,c^2 are \;in \;AP\)
(C)	\(\frac{a}{c}=\frac{sin(A-B)}{sin(B-C)}\)	3	\(\triangle\)
(D)	\(bc\;cos^2\frac{A}{2}\)	4	\(R\; r_1r_2r_3\)
		5	\(s(s-a)\)

• A. (A)-(4), (B)-(3), (C)-(1), (D)-(5)
• B. (A)-(5), (B)-(4), (C)-(3), (D)-(2)
• C. (A)-(3) , (B)-(1), (C)-(2) , (D)-(5)
• D. (A)-(4) , (B)-(5), (C)-(2) , (D)-(1)

Answer 1

The Correct Option is C

To solve the problem, we need to correctly match the expressions in List I with the descriptions in List II based on known mathematical identities and properties related to triangles. 

Let's analyze each expression:

1. \(r_1r_2\sqrt{\bigg(\frac{4R-r_1-r_2}{r_1+r_2}\bigg)}\): This expression is complex and appears to be related to geometric properties inside a triangle, such as the circumradius and inradius.
2. \(\frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}\): This expression involves the inradii of the triangle, which suggests a relationship linked to the area or specific angles in the triangle.
3. \(\frac{a}{c}=\frac{sin(A-B)}{sin(B-C)}\): In a triangle, the angles and sides are related by the law of sines, so this expression describes a specific condition about the sides \(a\) and \(c\), indicating their relationship in terms of angles \(A\) and \(B\).
4. \(bc\;cos^2\frac{A}{2}\): This expression involves the cosine of half an angle, which is often related to segment lengths or areas related to angle \(A\) in the triangle.

Next, let's explore what each number in List II refers to:

1. \(b\): Directly related to a side of the triangle, possibly connected with proportion or sine rules.
2. \(a^2, b^2, c^2 \text{ are in AP}\): This implies a specific arithmetic relationship among the squares of the sides of the triangle.
3. \(\triangle\): This symbol denotes the triangle area or a specific geometric property.
4. \(R \; r_1r_2r_3\): This refers to products involving the circumradius and inradii, possibly a complex relationship involving these radii.
5. \(s(s-a)\): This represents part of an expression related to the area of a triangle, especially in formulae such as Heron’s formula.

With this understanding, the correct matches are:

• (A) is matched with 3: The expression seems to be related to geometric properties linked with triangle area or similar computation.
• (B) is matched with 1: The involvement of inradii suggests the role these radii play in proportions, potentially linking to sine or cosine rules.
• (C) is matched with 2: This matches the condition of side lengths being proportional in accordance with specific angle calculations.
• (D) is matched with 5: Involving \(cos^2\frac{A}{2}\) and side lengths suggests a link to areas calculated using semi-perimeters.

Thus, the correct option is:

(A)-(3), (B)-(1), (C)-(2), (D)-(5)

.