A Republic is a form of government in which the head of state is elected (directly or indirectly) for a fixed term, rather than inheriting the position. Core principles include popular sovereignty, rule of law, and a written constitution.
In simple exam terms, a republic means that the legitimacy of the head of state comes from the people or elections, not hereditary succession. It does not define whether the executive system is presidential or parliamentary — both can exist within republics.
India’s Constitution establishes a Parliamentary Democracy and a Republic. The President of India—the head of state—is elected by an Electoral College, not by hereditary succession. This satisfies the republican principle of an elected head of state.
The Prime Minister heads the government, while the President holds a largely ceremonial role, symbolizing the unity of the Republic. Regular elections and constitutional term limits ensure no dynastic succession.
Brazil’s President is elected by direct popular vote and serves as both the head of state and head of government. The fixed-term tenure and absence of hereditary succession confirm its republican structure.
Additionally, Brazil’s federal system divides powers between the Union and states, but this territorial distribution does not affect its republican character.
Both India and Brazil derive authority from the consent of the people through elections. In both cases, the head of state is elected, not hereditary, fulfilling the core condition of a Republic.
\[ \textbf{Hence, India and Brazil are Republics because their Heads of State are elected by the people and not hereditary.} \]
Complete the following activity to prove that the sum of squares of diagonals of a rhombus is equal to the sum of the squares of the sides.
Given: PQRS is a rhombus. Diagonals PR and SQ intersect each other at point T.
To prove: PS\(^2\) + SR\(^2\) + QR\(^2\) + PQ\(^2\) = PR\(^2\) + QS\(^2\)
Activity: Diagonals of a rhombus bisect each other.
In \(\triangle\)PQS, PT is the median and in \(\triangle\)QRS, RT is the median.
\(\therefore\) by Apollonius theorem,
\[\begin{aligned} PQ^2 + PS^2 &= \boxed{\phantom{X}} + 2QT^2 \quad \dots \text{(I)} \\ QR^2 + SR^2 &= \boxed{\phantom{X}} + 2QT^2 \quad \dots \text{(II)} \\ \text{Adding (I) and (II),} \quad PQ^2 + PS^2 + QR^2 + SR^2 &= 2(PT^2 + \boxed{\phantom{X}}) + 4QT^2 \\ &= 2(PT^2 + \boxed{\phantom{X}}) + 4QT^2 \quad (\text{RT = PT}) \\ &= 4PT^2 + 4QT^2 \\ &= (\boxed{\phantom{X}})^2 + (2QT)^2 \\ \therefore \quad PQ^2 + PS^2 + QR^2 + SR^2 &= PR^2 + \boxed{\phantom{X}} \\ \end{aligned}\]