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3 years ago

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

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

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)

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Find the complex fourth roots of 81(cos(3pi/8) + i sin(3pi/8))

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By using De Moivre's theorem:

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 ⇒ z = 81(cos(3π/8) + i sin(3π/8))


Part (A) find the modulus for all of the fourth roots

∴ The modulus of the given complex number = l z l = 81

∴ 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

The first root = $z_{1} = 3 ( cos \ \frac{3\pi}{32} + i \ sin \ \frac{3\pi}{32})$
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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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

from this question

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