Question

Right triangle ABC and its image, triangle A'B'C' are shown in the image attached.

Algebraically prove that a clockwise and counterclockwise rotation of 180° about the origin for triangle ABC are equivalent rotations.

Answer 1

Triangle ABC ha vertices at: A(-3,6), B(0,-4) and (2,6).

Let us apply 90 degrees clockwise about the origin twice to obtain 180 degrees clockwise rotation.

We apply the 90 degrees clockwise rotation rule.

(x,y) --- (y, -x)

A(-3, 6) > (6, 3)

B(0, 4) > (4, 0)

C(2, 6) > (6, -2)

We apply the 90 degrees clockwise rotation rule again on the resulting points:

(6, 3) > A'(-3, 6)

(4, 0) > B'(0, -4)

(6, -2)> C'(-2, -6)

Let us now apply 90 degrees counterclockwise rotation about the origin twice to obtain 180 degrees counterclockwise rotation.

We apply the 90 degrees counterclockwise rotation rule.

(x,y) --- (-y, x)

A(-3, 6) > (-6, -3)

B(0, 4) > (-4, 0)

C(2, 6) > (-6, 2)

We apply the 90 degrees counterclockwise rotation rule again on the resulting points:

(-6, -3) > A'(3, -6)

(-4, 0) > B'(0, -4)

(-6, 2) > C'(-2, -6)

We can see that A'(3,-6), B'(0,-4) and C'(-2,-6) is the same for both the 180 degrees clockwise and counterclockwise rotations.

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Answer 2

See explanation

Explanation:

Triangle ABC ha vertices at: A(-3,6), B(0,-4) and (2,6).

Let us apply 90 degrees clockwise about the origin twice to obtain 180 degrees clockwise rotation.

We apply the 90 degrees clockwise rotation rule.

`(x,y)\to (y,-x)`

`\implies A(-3,6)\to (6,3)`

`\implies B(0,4)\to (4,0)`

`\implies C(2,6)\to (6,-2)`

We apply the 90 degrees clockwise rotation rule again on the resulting points:

`\implies (6,3)\to A''(3,-6)`

`\implies (4,0)\to B''(0,-4)`

`\implies (6,-2)\to C''(-2,-6)`

Let us now apply 90 degrees counterclockwise rotation about the origin twice to obtain 180 degrees counterclockwise rotation.

We apply the 90 degrees counterclockwise rotation rule.

`(x,y)\to (-y,x)`

`\implies A(-3,6)\to (-6,-3)`

`\implies B(0,4)\to (-4,0)`

`\implies C(2,6)\to (-6,2)`

We apply the 90 degrees counterclockwise rotation rule again on the resulting points:

`\implies (-6,-3)\to A''(3,-6)`

`\implies (-4,0)\to B''(0,-4)`

`\implies (-6,2)\to C''(-2,-6)`

We can see that A''(3,-6), B''(0,-4) and C''(-2,-6) is the same for both the 180 degrees clockwise and counterclockwise rotations.