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Nikolay [14]
3 years ago
5

2. Consider polygon ABCD with coordinates A(8,12), B(10,4),C(4,4),D(2,8).

Mathematics
2 answers:
max2010maxim [7]3 years ago
4 0

Answer:

Step-by-step explanation:

Lets take point A:

Reflection across the line y = 2 moves the point to

(8, -8)

A dilation of 1/2 will bring the above point to (1/2*8, 1/28-8= (4, -4).

So for the other points we have

B(10,4)-->B'(10, 0)---> B"( 5, 0)

C(4, 4)--> C'(4, 0) ------C"(2, 0)

D(2, 8)----D'(2, -4)---->D"(1, -2)

lina2011 [118]3 years ago
3 0

<u>Find the coordinates after reflection: A′(8,−8)andB′(10,0)andC′(4,0)andD′(2,−4)</u>

<u>Find the coordinates after reflection: A′(8,−8)andB′(10,0)andC′(4,0)andD′(2,−4)Find the coordinates after dilation: A″(4,−4)andB″(5,0)andC″(2,0)andD″(1,−2)</u>

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BartSMP [9]
By using <span>De Moivre's theorem:
</span>
If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴ \sqrt[n]{z} =  \sqrt[n]{a} \ (cos \  \frac{\theta + 360K}{n} + i \ sin \ \frac{\theta +360k}{n} )
k= 0, 1 , 2, ..... , (n-1)


For The given complex number <span>⇒ z = 81(cos(3π/8) + i sin(3π/8))
</span>

Part (A) <span>find the modulus for all of the fourth roots
</span>
<span>∴ The modulus of the given complex number = l z l = 81
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∴ The modulus of the fourth root = \sqrt[4]{z} =  \sqrt[4]{81} = 3

Part (b) find the angle for each of the four roots

The angle of the given complex number = \frac{3 \pi}{8}
There is four roots and the angle between each root = \frac{2 \pi}{4} =  \frac{\pi}{2}
The angle of the first root = \frac{ \frac{3 \pi}{8} }{4} =  \frac{3 \pi}{32}
The angle of the second root = \frac{3\pi}{32} +  \frac{\pi}{2} =  \frac{19\pi}{32}
The angle of the third root = \frac{19\pi}{32} +  \frac{\pi}{2} =  \frac{35\pi}{32}
The angle of the  fourth root = \frac{35\pi}{32} +  \frac{\pi}{2} =  \frac{51\pi}{32}

Part (C): find all of the fourth roots of this

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The second root = z_{2} = 3 ( cos \  \frac{19\pi}{32} + i \ sin \ \frac{19\pi}{32})

The third root = z_{3} = 3 ( cos \  \frac{35\pi}{32} + i \ sin \ \frac{35\pi}{32})
The fourth root = z_{4} = 3 ( cos \  \frac{51\pi}{32} + i \ sin \ \frac{51\pi}{32})
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from this question

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