The rectangle R’S’T’U’ is a 90° counterclockwise rotation of RSTU so options (A) and (C) will be correct.
<h3>What is a rectangle?</h3>
The rectangle is a geometrical figure in which opposite sides are equal.
The angle between any two consecutive sides will be 90 degrees.
Area of rectangle = length × width.
Perimeter of rectangle = 2( length + width).
Given the rectangle RSTU if we plot it then it becomes a major horizontal rectangle.
If we transpose into R’S’T’U’ then it rotates 90 degrees counterclockwise and becomes a major vertical ractangle.
Hence the rectangle RSTU will rotate 90° counterclockwise to form R’S’T’U’.
For more about rectangles,
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Answer:
![x = 25.9](https://tex.z-dn.net/?f=x%20%3D%2025.9)
Step-by-step explanation:
Apply law of sines.
![\frac{23}{ \sin(22) } = \frac{x}{ \sin(25) }](https://tex.z-dn.net/?f=%20%5Cfrac%7B23%7D%7B%20%5Csin%2822%29%20%7D%20%20%3D%20%20%5Cfrac%7Bx%7D%7B%20%5Csin%2825%29%20%7D%20)
![\frac{23}{ \sin(22) } \times \sin(25) = x](https://tex.z-dn.net/?f=%20%5Cfrac%7B23%7D%7B%20%5Csin%2822%29%20%7D%20%20%5Ctimes%20%20%5Csin%2825%29%20%20%3D%20x)
![x = 25.9](https://tex.z-dn.net/?f=x%20%3D%2025.9)
Answer:
![\frac{1}{2\sqrt{5} }](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%5Csqrt%7B5%7D%20%7D)
Step-by-step explanation:
Let,
= y
sin(y) = ![\frac{x}{6}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B6%7D)
![\frac{d}{dx}\text{sin(y)}=\frac{d}{dx}(\frac{x}{6})](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Ctext%7Bsin%28y%29%7D%3D%5Cfrac%7Bd%7D%7Bdx%7D%28%5Cfrac%7Bx%7D%7B6%7D%29)
![\frac{d}{dx}\text{sin(y)}=\frac{1}{6}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Ctext%7Bsin%28y%29%7D%3D%5Cfrac%7B1%7D%7B6%7D)
---------(1)
![\frac{1}{6}=\text{cos}(y)\frac{dy}{dx}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B6%7D%3D%5Ctext%7Bcos%7D%28y%29%5Cfrac%7Bdy%7D%7Bdx%7D)
![\frac{dy}{dx}=\frac{1}{6\text{cos(y)}}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B1%7D%7B6%5Ctext%7Bcos%28y%29%7D%7D)
cos(y) = ![\sqrt{1-\text{sin}^{2}(y) }](https://tex.z-dn.net/?f=%5Csqrt%7B1-%5Ctext%7Bsin%7D%5E%7B2%7D%28y%29%20%7D)
= ![\sqrt{1-(\frac{x}{6})^2}](https://tex.z-dn.net/?f=%5Csqrt%7B1-%28%5Cfrac%7Bx%7D%7B6%7D%29%5E2%7D)
= ![\sqrt{1-(\frac{x^2}{36})}](https://tex.z-dn.net/?f=%5Csqrt%7B1-%28%5Cfrac%7Bx%5E2%7D%7B36%7D%29%7D)
Therefore, from equation (1),
![\frac{dy}{dx}=\frac{1}{6\sqrt{1-\frac{x^2}{36}}}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B1-%5Cfrac%7Bx%5E2%7D%7B36%7D%7D%7D)
Or ![\frac{d}{dx}[\text{sin}^{-1}(\frac{x}{6})]=\frac{1}{6\sqrt{1-\frac{x^2}{36}}}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Ctext%7Bsin%7D%5E%7B-1%7D%28%5Cfrac%7Bx%7D%7B6%7D%29%5D%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B1-%5Cfrac%7Bx%5E2%7D%7B36%7D%7D%7D)
At x = 4,
![\frac{d}{dx}[\text{sin}^{-1}(\frac{4}{6})]=\frac{1}{6\sqrt{1-\frac{4^2}{36}}}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Ctext%7Bsin%7D%5E%7B-1%7D%28%5Cfrac%7B4%7D%7B6%7D%29%5D%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B1-%5Cfrac%7B4%5E2%7D%7B36%7D%7D%7D)
![\frac{d}{dx}[\text{sin}^{-1}(\frac{2}{3})]=\frac{1}{6\sqrt{1-\frac{16}{36}}}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Ctext%7Bsin%7D%5E%7B-1%7D%28%5Cfrac%7B2%7D%7B3%7D%29%5D%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B1-%5Cfrac%7B16%7D%7B36%7D%7D%7D)
![=\frac{1}{6\sqrt{\frac{36-16}{36}}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B%5Cfrac%7B36-16%7D%7B36%7D%7D%7D)
![=\frac{1}{6\sqrt{\frac{20}{36} }}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B6%5Csqrt%7B%5Cfrac%7B20%7D%7B36%7D%20%7D%7D)
![=\frac{1}{\sqrt{20}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B20%7D%7D)
![=\frac{1}{2\sqrt{5}}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%5Csqrt%7B5%7D%7D)
there are 7 verticles on this triangular prism