Area of a Triangle
The area of a polygon is the number of square units inside that polygon.The area is 2-dimensional like a carpet or an area rug. A triangle is a three-sided polygon.
Summary
The area of a polygon is the number of square units inside that polygon.The area is 2-dimensional like a carpet or an area rug. A triangle is a three-sided polygon.
Things to Remember
\( Area \: of \: triangle = \frac{1}{2} \times base \times altitude. \)
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Subjective Questions
Q1:
What are fundamental rights?
Type: Short Difficulty: Easy
Q2:
What are the importance of fundamental rights? Why do people need them?
Type: Short Difficulty: Easy
Q3:
What kinds of rights were passed by the worldwide declaration of human rights?
Type: Long Difficulty: Easy
<ul>
<li>Everyone has the right to life and security of person.</li>
<li>Everyone has the right to follow a religion of one's choice.</li>
<li>Everyone has the right to recognition everywhere as a person before the law.</li>
<li>Everyone has the right to freedom of movement within the borders of each state.</li>
<li>Everyone has the right to leave any country, including his own, and to return to his country.</li>
<li>Everyone has right to earn.</li>
<li>Everyone has the right to freedom of peaceful assembly and association.</li>
<li>Everyone has the right to work, in favourable conditions of work and to protection against unemployment.</li>
</ul>
Q4:
What is regarded illegal under right to equality?
Type: Short Difficulty: Easy
<p> </p>
Q5:
What kinds of freedom do citizens have?
Type: Short Difficulty: Easy
Q6:
List the fundamental rights.
Type: Short Difficulty: Easy
<ul>
<li>Rights to privacy</li>
<li>Rights regarding labour</li>
<li>Rights against torture</li>
<li>Rights against exile</li>
<li>Rights to social justice</li>
<li>Rights against preventive detention</li>
<li>Rights of women</li>
<li>Rights of Dalits</li>
<li>Right to health care</li>
<li>Rights regarding environment and health</li>
</ul>
Q7:
Write any two important fundamental rights?
Type: Short Difficulty: Easy
<ul>
<li>Right to equality</li>
<li>Right to freedom</li>
</ul>
<p> </p>
<p> </p>
Q8:
List the fundamental rights of children.
Type: Short Difficulty: Easy
<ul>
<li>Rights to privacy</li>
<li>Rights against torture</li>
<li>Rights regarding environments and health</li>
<li>Rights against exile</li>
<li>Rights of women</li>
<li>Rights against preventive detention</li>
</ul>
Q9:
Write any two declarations of human rights passed by UNO ?
Type: Short Difficulty: Easy
<p>Everyone has the right to live and security of person.</p>
<p>Everyone has the right to follow a religion of one's choice.</p>
Q10:
Write down any four rights of children.
Type: Very_short Difficulty: Easy
<ul>
<li>Right to read</li>
<li>Right to play</li>
<li>Right to think</li>
<li>Right to eat</li>
</ul>
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Fundamental right
RIght to equality
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Rights of children
Fundamental rights
Area of a Triangle
We already know that
\( Area \: of \: triangle = \frac{1}{2} \times base \times altitude. \)
In the \( \Delta ABC \) , side BC is the base and AD is its height,
Now,
\begin{align*} Area \: of \: \Delta ABC &= \frac{1}{2} \times base \times height \\ &= \frac{1}{2} \times BC \times AD\\ &= \frac{1}{2} bh \: (\because BC = b \: and \: AD = h )\end{align*}
If three sides of a triangle a, b and c are known,
area of a triangle is obtained as,
$$ \Delta = \sqrt{s( s - a ) ( s - b) ( s - c )} $$
where, \(S = \frac{a + b + c}{2}\) (semi-perimeter of triangle)
Here, We can find the area of a triangle by another formula. If two sides and angle included between them is given.
\begin{align*} Area \: of \: \Delta ABC &= \frac{1}{2} \times base \times height \\ &= \frac{1}{2} \times BC \times AD\\ &= \frac{1}{2} \times a \times h \: \: \: ........ (1)\end{align*}
From triangle ADB,
\begin{align*} Sin \: B &= \frac{AD}{AB}\\ &= \frac{h}{c}\\ or, h &= c \: sin \: B \end{align*}
Substituting the value of h in(1) we get,
\begin{align*} Area \: of \: \Delta ABC &= \frac{1}{2} \times a \times c \: Sin \: B \end{align*}
\(\boxed{\therefore Area \: of \Delta ABC = \frac{1}{2} \: ac \: sin \: B}\)
In the above triangle ABC, \(\angle B \) is acute. The above formula is valid even if \(\angle B\) is obtuse.
Similarly,
When two sides b, c and \(\angle A\) of \(\Delta ABC\) are given then,
\(\boxed{\therefore Area \: of \Delta ABC = \frac{1}{2} \: bc \: sin \: A}\)
and when two sides a, b and \(\angle C \) are given then,
\(\boxed{\therefore Area \: of \Delta ABC = \frac{1}{2} \: ab \: sin \: C}\)
Since \(\frac{1}{2}\) ab sin C, \(\frac{1}{2}\) bc sin A and \(\frac{1}{2}\) ac sin B represent the area of same triangle ABC.
So,
\begin{align*} \frac{1}{2} ab \: sin \: C &= \frac{1}{2} bc \: sin \: A = \frac{1}{2} ac \: sin \: B \\ \frac{\frac{1}{2} ab \: sin \: C}{\frac{1}{2} abc}&= \frac{\frac{1}{2}bc \: sin \: A}{\frac{1}{2}abc} = \frac{\frac{1}{2}ac \: sin \: B}{\frac{1}{2}abc}\\ or, \frac{sin \: C}{c} &= \frac{sin \: A}{a} = \frac{sin \: B}{b}\\ or, \frac{a}{sinA} &= \frac{b}{sin B} = \frac{c}{sinC}= constant\end{align*}
This is called the sine law of trigonometry.
Lesson
Trignometry
Subject
Compulsory Mathematics
Grade
Grade 10
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