Math / Geometry

QuestionAn equilateral triangle is shown below. Line $m$ passes through a vertex and bisects a side. Line $n$ bisects each side it passes through. Point $Y$ is the center of the triangle.
Which transformation(s) must map the triangle exactly onto itself? Choose all that apply. Counterclockwise rotation about $Y$ by $120^{\circ}$ Reflection across line $m$ Clockwise rotation about $Y$ by $360^{\circ}$ Reflection across line $n$ None of the above

Studdy Solution

STEP 1

1. The triangle is equilateral, meaning all sides and angles are equal.
2. Point $Y$ is the centroid, which is also the center of rotation for the triangle.
3. Line $m$ passes through a vertex and bisects the opposite side.
4. Line $n$ bisects each side it passes through.

STEP 2

1. Analyze the effect of a counterclockwise rotation about $Y$ by $120^{\circ}$ .
2. Analyze the effect of a reflection across line $m$ .
3. Analyze the effect of a clockwise rotation about $Y$ by $360^{\circ}$ .
4. Analyze the effect of a reflection across line $n$ .
5. Determine if any of the transformations map the triangle onto itself.

STEP 3

Analyze the effect of a counterclockwise rotation about $Y$ by $120^{\circ}$ :
- In an equilateral triangle, a $120^{\circ}$ rotation about the centroid $Y$ will map each vertex to the position of the next vertex in a cyclic manner. - Therefore, this transformation will map the triangle onto itself.

STEP 4

Analyze the effect of a reflection across line $m$ :
- Since line $m$ passes through a vertex and bisects the opposite side, it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 5

Analyze the effect of a clockwise rotation about $Y$ by $360^{\circ}$ :
- A $360^{\circ}$ rotation is a full rotation, meaning the triangle will map onto itself as it returns to its original position.

STEP 6

Analyze the effect of a reflection across line $n$ :
- Line $n$ bisects each side it passes through, which implies it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 7

Determine if any of the transformations map the triangle onto itself:
- Counterclockwise rotation about $Y$ by $120^{\circ}$ maps the triangle onto itself. - Reflection across line $m$ maps the triangle onto itself. - Clockwise rotation about $Y$ by $360^{\circ}$ maps the triangle onto itself. - Reflection across line $n$ maps the triangle onto itself.
The transformations that map the triangle exactly onto itself are: - Counterclockwise rotation about $Y$ by $120^{\circ}$ - Reflection across line $m$ - Clockwise rotation about $Y$ by $360^{\circ}$ - Reflection across line $n$

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Studdy Solution

STEP 1

1. The triangle is equilateral, meaning all sides and angles are equal.
2. Point $Y$ is the centroid, which is also the center of rotation for the triangle.
3. Line $m$ passes through a vertex and bisects the opposite side.
4. Line $n$ bisects each side it passes through.

STEP 2

STEP 3

Analyze the effect of a counterclockwise rotation about $Y$ by $120^{\circ}$ :
- In an equilateral triangle, a $120^{\circ}$ rotation about the centroid $Y$ will map each vertex to the position of the next vertex in a cyclic manner. - Therefore, this transformation will map the triangle onto itself.

STEP 4

Analyze the effect of a reflection across line $m$ :
- Since line $m$ passes through a vertex and bisects the opposite side, it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 5

Analyze the effect of a clockwise rotation about $Y$ by $360^{\circ}$ :
- A $360^{\circ}$ rotation is a full rotation, meaning the triangle will map onto itself as it returns to its original position.

STEP 6

Analyze the effect of a reflection across line $n$ :
- Line $n$ bisects each side it passes through, which implies it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 7

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Studdy Solution

STEP 1

1. The triangle is equilateral, meaning all sides and angles are equal.2. Point Y Y Y is the centroid, which is also the center of rotation for the triangle.3. Line m m m passes through a vertex and bisects the opposite side.4. Line n n n bisects each side it passes through.

STEP 2

STEP 3

STEP 4

Analyze the effect of a reflection across line m m m:- Since line m m m passes through a vertex and bisects the opposite side, it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 5

Analyze the effect of a clockwise rotation about Y Y Y by 360∘ 360^{\circ} 360∘:- A 360∘ 360^{\circ} 360∘ rotation is a full rotation, meaning the triangle will map onto itself as it returns to its original position.

STEP 6

Analyze the effect of a reflection across line n n n:- Line n n n bisects each side it passes through, which implies it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The triangle is equilateral, meaning all sides and angles are equal.
2. Point $Y$ is the centroid, which is also the center of rotation for the triangle.
3. Line $m$ passes through a vertex and bisects the opposite side.
4. Line $n$ bisects each side it passes through.

Analyze the effect of a reflection across line $m$ :
- Since line $m$ passes through a vertex and bisects the opposite side, it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.

Analyze the effect of a clockwise rotation about $Y$ by $360^{\circ}$ :
- A $360^{\circ}$ rotation is a full rotation, meaning the triangle will map onto itself as it returns to its original position.

Analyze the effect of a reflection across line $n$ :
- Line $n$ bisects each side it passes through, which implies it is an axis of symmetry for the equilateral triangle. - Reflecting the triangle across this line will map the triangle onto itself.