Lizard – Rotation and Reflection

Lizard – Rotation and Reflection
1) Rotate the lizard about the point 𝑅 (12,7) through 90°.

VECTOR (From R to the point)

VECTOR coordinates (from the point 𝑅 (12,7) to each point)

Rotation of 90°

(𝑥,𝑦) →(−𝑦,𝑥)

Final actual image coordinates

𝑅𝐴̅̅̅̅

(0,5)

(−5,0)

𝐴′ (7,7)

𝑅𝐵̅̅̅̅

(3,5)

(−5,3)

𝐵′ (7,10)

𝑅𝐶̅̅̅̅

(5,5)

(−5,5)

𝐶′ (7,12)

𝑅𝐷̅̅̅̅

(7,3)

(−3,7)

𝐷′ (9 ,14)

𝑅𝐸̅̅̅̅

(7,7)

(−7,7)

𝐸′ (5,14)

𝑅𝐹̅̅̅̅

(8,6)

(−6,8)

𝐹′ (6,15)

𝑅𝐺̅̅̅̅

(5,8)

(−8,5)

𝐺′ (4,12)

𝑅𝐻̅̅̅̅

(2,9)

(−9,2)

𝐻′ (3,9)
1) Rotate the lizard about the point 𝑅 (12,7) through 90°.
2) Reflect the lizard across the line

y=12x+16y = \frac{1}{2}x + 16

 

First, rewrite the equation in the form:

Ax1+By1+C=0

The equation given is:

y=12x+16y = \frac{1}{2}x + 16

Multiply both sides by 2:

2y=x+322y = x + 32

Rewrite in standard form (as above):

x – 2y + 32 = 0

Screenshot 2024 10 18 at 11.23.30 PM

In the equation above (in blue), we have:

A=1,B=−2,C=32A = 1, \quad B = -2, \quad C = 32

To find the

xx

-value of the reflected point of each given point, we use the following equations:

1) Rotate the lizard about the point 𝑅 (12,7) through 90°.

Substituting the values of 𝐴 = 1, 𝐵 = −2, 𝐶 = 32, we get:

1) Rotate the lizard about the point 𝑅 (12,7) through 90°.

We now find the reflected point for each point on the lizard:

Let’s look at the first example: Point 𝐴(12,12). This point has the coordinates Screenshot 2024 10 18 at 11.38.15 PM

𝑥 value of the reflected point

𝑦 value of the reflected point

x−12=−2(1(12)−2(12)+32)1+4x – 12 = \frac{-2(1(12) – 2(12) + 32)}{1 + 4} x−12=−2(12−24+32)5x – 12 = \frac{-2(12 – 24 + 32)}{5} x−12=−2(20)5x – 12 = \frac{-2(20)}{5} x−12=−8x – 12 = -8 
x=4x = 4 

y−12=−2(1(12)−2(12)+32)−2(1+4)y – 12 = \frac{-2(1(12) – 2(12) + 32)}{-2(1 + 4)} y−12=−2(12−24+32)−2(5)y – 12 = \frac{-2(12 – 24 + 32)}{-2(5)} y−12=−2(20)−2(5)y – 12 = \frac{-2(20)}{-2(5)} y−12=−8y – 12 = -8 
y=28y = 28 

So the coordinates of 𝐴 are (12,12) and the coordinates of 𝐴’ are (4,28)

Let’s now perform the same operations on all the other points

Screenshot 2024 10 18 at 11.42.55 PM
Screenshot 2024 10 18 at 11.43.25 PM
Screenshot 2024 10 18 at 11.44.40 PM
Screenshot 2024 10 18 at 11.45.33 PM
Screenshot 2024 10 18 at 11.46.02 PM

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