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what is the length of the hypotenuse of a right triangle with; what is the area of the shaded sector of the circle; what is the length of the hypotenuse of a 30 60 90

Sagot :

The length of the hypotenuse of the right-angled triangle = is 60 units

The area of the shaded region of the circle = 900[tex]\pi[/tex] - 450[tex]\sqrt{3}[/tex] = 2048.01 square units.

According to the attached figure,

Let the area of the large part i.e area of the circle

⇒ [tex]A_{c}[/tex] = [tex]\pi r^{2}[/tex]

Let the area of the triangle,

⇒ [tex]A_{t}[/tex] = [tex]\frac{1}{2}[/tex] x base x height.

Let the height of the triangle be 30 units

So according to the special feature of a 30° 60° 90° degrees triangle

∴ The length of the hypotenuse is twice that of the length of the height of the right-angled triangle and the length of the base is the [tex]\sqrt{3}[/tex] multiple of the height of the triangle.

So the length of the hypotenuse = 2 x 30 =60 units

The length of the base of the triangle= 30[tex]\sqrt{3}[/tex]

Now putting the value of lengths in the formula of the area of the triangle,

⇒ [tex]A_{t}[/tex] = [tex]\frac{1}{2}[/tex] x base x height

⇒ [tex]A_{t}[/tex] = [tex]\frac{1}{2}[/tex] x 30[tex]\sqrt{3}[/tex] x 30

⇒ [tex]A_{t}[/tex] = 450[tex]\sqrt{3}[/tex] sq units

Since the right-angled triangle is carved into the circle, the length of the hypotenuse is equal to the diameter of the circle and the radius of the circle is half of the diameter,

Diameter of circle = 60 units

Radius ( r ) = 20/2 = 30 units

Putting the values in the area of the circle,

⇒ [tex]A_{c}[/tex] = [tex]\pi r^{2}[/tex]

⇒ [tex]A_{c}[/tex] = [tex]\pi[/tex] x 30 x 30

⇒ [tex]A_{c}[/tex] = 900[tex]\pi[/tex] sq units

Now to calculate the area of the shaded region we need to subtract the smallest area ( area of the triangle ) from the greater area ( area of the circle ),

Let the area of the shaded portion ( [tex]A_{s}[/tex] ) = [tex]A_{c}[/tex] - [tex]A_{t}[/tex]

[tex]A_{s}[/tex] = 900[tex]\pi[/tex] - 450[tex]\sqrt{3}[/tex]

[tex]A_{s}[/tex] = 2048.01 sq units

Therefore, the length of the hypotenuse = is 60 units, and the area of the shaded portion = is 2048.01 sq units

To learn more about Area of Circles,

https://brainly.com/question/14989383

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