Question

the perimeter of right triangle is 30 cm its hypotenuse is 13 cm and base is 5 cm. Find the area of the triangle​

Answer 1

Solution :

$\bf{\large{\underline{\bf{Given\::}}}}}$

The perimeter of a right triangle is 30 cm, It's hypotenuse is 13 cm and base is 5 cm.

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\thicklines\put(7.7,2.9){\large{A}}\put(7.7,1){\large{B}}\put(10.6,1){\large{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\put(10.5,1){\line(-4,3){2.5}}\put(7.2,2){\sf{\large{r\:cm}}}\put(9,0.7){\sf{\large{5\:cm}}}\put(9.4,1.9){\sf{\large{13\:cm}}}\put(8.2,1){\line(1,0){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

$\bf{\large{\underline{\bf{To\:find\::}}}}}$

The area of the triangle.

$\bf{\large{\underline{\bf{Explanation\::}}}}}$

Let the height of the right angled triangle be r cm

$\bf{We\:have}\begin{cases}\sf{The\:Perimeter\:of\:\triangle=30\:cm}\\ \sf{The\:Hypotenuse\:(H)=13\:cm}\\ \sf{The\:base\:(B)=5\:cm}\end{cases}}$

$\bf{\large{\orange{\underline{\underline{\tt{Using\:Pythagoras\:Theorem\::}}}}}}$

$\longrightarrow\underbrace{\sf{(Hypotenuse)^{2}} =(Base)^{2} +(Perpendicular)^{2}}\\\\\longrightarrow\sf{(13cm)^{2} =(5cm)^{2} +(r)^{2} }\\\\\longrightarrow\sf{169cm^{2} =25cm^{2} +(r)^{2} }\\\\\longrightarrow\sf{(r)^{2} =169cm^{2}-25cm^{2} }\\\\\longrightarrow\sf{(r)^{2} =144cm^{2} }\\\\\longrightarrow\sf{r=\sqrt{144cm^{2} } }\\\\\longrightarrow\sf{\red{r=12\:cm}}$

∴ The height of the right angled triangle is 12 cm

We know that formula of the area of right angled Δ :

$\longrightarrow\sf{Area\:_{triangle}=\dfrac{1}{2} \times base\times height}\\\\\longrightarrow\sf{Area\:_{triangle}=\dfrac{1}{\cancel{2}} \times 5cm\times \cancel{12cm}}\\\\\longrightarrow\sf{Area\:_{triangle}=(5\times 6)cm^{2} }\\\\\longrightarrow\sf{\red{Area\:_{triangle}=30\:cm^{2} }}$

Thus;

$\underline{\sf{The\:area\:of\:right\:angled\:triangle\:is\:\:30cm^{2} }}}$

Answer 2

GIVEN:

Perimeter of right traingle = 30cm

Two sides of the traingle = 13cm and 5cm

TO FIND:

Area of the triangle

SOLUTION:

Let the third side be "x"

We know that,

Perimeter of triangle = Sum of all sides

==> x + 13 + 5 = 30

==> x + 18 = 30

==> x = 30 - 18

==> x = 12cm

Opposite = Height = 12cm

Area of right angle triangle = 1/2 × base × height

==> 1/2 × 5 × 12

==> 1/2 × 60

==> 30cm²

Therefore, the area of triangle is 30cm².