Verified answer

Answer:

>>> 1) 16√3 cm²

>>> 2) 30 cm²

Step-by-step explanation:

#Question 1 :-

Given that ,

• Perimeter of equilateral triangle is 24 cm

We have to find ,

• Area of equilateral triangle

Solution :

Finding Length of equilateral triangle using its Perimeter.

[tex] \: \: \: \boxed{\sf{Perimeter_{(Equilateral \: Triangle)} = 3 \times a }}[/tex]

Where ,

a refers to length/side of triangle

[tex] \longmapsto \: \: \: \sf{3 \times a = 24}[/tex]

Dividing both sides with 3 :

[tex]\longmapsto \: \: \: \sf{ \dfrac{( \cancel{3} \times a )}{ \cancel{3}}= \cancel{\dfrac{ 24}{3}}} [/tex]

We get ,

[tex]\longmapsto \: \: \: \sf{ \bold{a = 8 \: cm}}[/tex]

• Therefore , Length of equilateral triangle is 8 cm

Now , Finding area of equilateral triangle :

[tex] \: \: \: \: \: \: \: \boxed{\sf{Area_{(Equilateral \: Triangle)} = \dfrac{ \sqrt{3} \: a {}^{2} }{4} }}[/tex]

Where ,

• a is the length/side of equilateral triangle

[tex] \dashrightarrow \: \: \: \sf{ \dfrac{ \sqrt{3} \: \times {8}^{2} }{4}}[/tex]

Simplifying :

[tex]\dashrightarrow \: \: \: \sf{ \dfrac{ \sqrt{3} \: \times \cancel{64}}{ \cancel{4}}}[/tex]

[tex]\dashrightarrow \: \: \: \underline{ \boxed{ \sf{ \bold{16 \sqrt{3} \: cm {}^{2} }}}} \: \: \: \bigstar[/tex]

• Therefore, area of an equilateral triangle is "16√3" cm²

#Question 2 :-

Given Right Angle Triangle ,

• Whose Hypotenuse is 13 cm

• Length of side is 5 cm

We have to find ,

• Area of right angled triangle

Solution :

Let us say that ,

• Third side of triangle is x cm

For finding area of triangle we need to find the length of its third side using Pythagoras Theorem :

[tex] \longmapsto \: \: \: \sf{x {}^{2} + {5}^{2} = 13 {}^{2} }[/tex]

[tex]\longmapsto \: \: \: \sf{x {}^{2} + 25 = 169 }[/tex]

Transposing 25 to RHS :

[tex]\longmapsto \: \: \: \sf{x {}^{2} = 169 - 25 }[/tex]

[tex]\longmapsto \: \: \: \sf{x {}^{2} = 144 }[/tex]

[tex]\longmapsto \: \: \: \sf{x = \sqrt{144} }[/tex]

[tex]\longmapsto \: \: \: \sf{ \bold{x = 12 \: cm }}[/tex]

• Therefore , third side of triangle is of 12 cm

Now, finding area of right angled triangle :

[tex] \: \: \boxed{\sf{Area_{(Right \: Angled \: Triangle)} = \dfrac{1}{2} \times (ab)}}[/tex]

Where ,

• a and b refers to sides of triangle other than hypotenuse

[tex] \dashrightarrow \: \: \: \sf{ \dfrac{1}{ \cancel{2}} \times ( \cancel{12 }\times 5) }[/tex]

[tex] \dashrightarrow \: \: \: \sf{6 \times 5}[/tex]

We get ,

[tex]\dashrightarrow \: \: \: \underline{ \boxed{ \sf{ \bold{30 \: cm {}^{2} }}}} \: \: \: \bigstar[/tex]

• Therefore , area of right angled triangle is "30" cm²