Lesson Worksheet: Congruence of Polygons through Transformations Mathematics • 8th Grade

In this worksheet, we will practice identifying congruence of polygons after applying transformations.

Q1:

A triangle 𝐴𝐡𝐢 has been dilated from a center 𝑃 by a scale factor of 3 to triangle 𝐴𝐡𝐢.

Are triangles 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′ similar?

  • ANo
  • BYes

Are triangles 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′ congruent?

  • AYes
  • BNo

Q2:

Determine, by applying transformations, whether the two triangles seen in the given figure are congruent.

  • AThey are not congruent.
  • BThey are congruent.

Q3:

If there exists a combination of rotations, reflections, and translations that would map one shape to another, would the two shapes be congruent?

  • ANo
  • BYes

Q4:

If triangle 𝐴 is mapped by a reflection in the line 𝑦=π‘₯ to triangle 𝐴′, would the two triangles be congruent?

  • AYes
  • BNo

Q5:

If triangle 𝐡 is mapped by a 180∘ rotation about the origin to triangle 𝐡′, would the two triangles be congruent?

  • AYes
  • BNo

Q6:

A triangle 𝐴𝐡𝐢 is rotated by 180∘ about the origin to triangle 𝐴𝐡𝐢.

Are triangles 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′ similar?

  • AYes
  • BNo

Are triangles 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′ congruent?

  • AYes
  • BNo

Q7:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴𝐡𝐢, which has then been transformed onto triangle 𝐴𝐡𝐢, as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • AA translation two right and three up
  • BA translation two left and three down
  • CA translation three right and two down
  • DA translation two left and three up
  • EA translation two right and three down

Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • AA translation four right and one down
  • BA rotation of 90∘ counterclockwise about point 𝐹
  • CA translation one right and four down
  • DA rotation of 90∘ clockwise about point 𝐸
  • EA reflection in the line ⃖⃗𝐸𝐹

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ congruent?

  • AYes
  • BNo

Q8:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′ as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • Aa translation three left and two up
  • Ba translation two right and three up
  • Ca translation three right and two up
  • Da translation three right and two down
  • Ea translation three left and two down

Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • Aa rotation of 90∘ counterclockwise about 𝐸
  • Ba translation four right
  • Ca rotation of 90∘ counterclockwise about 𝐹
  • Da rotation of 90∘ clockwise about 𝐹
  • Ea reflection in the line ⃖⃗𝐸𝐹

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ congruent?

  • ANo
  • BYes

Q9:

The figure shows triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • ANo
  • BYes

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐡𝐢 can be reflected to obtain triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • BNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐡𝐢 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.
  • CTriangle 𝐴𝐡𝐢 can be rotated to obtain triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • DWe can apply a two-stage transformation on triangle 𝐴𝐡𝐢 involving a reflection and then a rotation to obtain triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.

Q10:

The figure shows triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐹.

Are the two triangles congruent?

  • AYes
  • BNo

Justify your answer with one of the following reasons.

  • AWe can apply a two-stage transformation on triangle 𝐴𝐡𝐢 involving a reflection and then a rotation to obtain triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • BTriangle 𝐴𝐡𝐢 can be rotated to obtain triangle 𝐷𝐸𝐹 and, thus, the triangles are congruent.
  • CNo sequence of translations, reflections, or rotations exists that can map triangle 𝐴𝐡𝐢 onto triangle 𝐷𝐸𝐹 and, therefore, the two triangles cannot be congruent.

Q11:

If triangle 𝑇 is mapped to triangle π‘‡οŽ˜ by a reflection, translation, or rotation, which of the following statements will be true of the two triangles?

  • AThey will be similar.
  • BThey are the same.
  • CThey have exactly one side of the same length.
  • DThey will be congruent.

Q12:

Triangle 𝐴𝐡𝐢 has been reflected in the line 𝐿 to obtain triangle 𝐴𝐡𝐢 as seen in the given figure.

Are the corresponding angles and sides of the two triangles equal?

  • ANo
  • BYes

What is the length of 𝐡𝐢?

What is the length of 𝐴′𝐡′?

What is the perimeter of triangle 𝐴𝐡𝐢?

Q13:

The triangle 𝐴𝐡𝐢 has been transformed onto triangle 𝐴′𝐡′𝐢′ which has then been transformed onto triangle 𝐴′′𝐡′′𝐢′′ as seen in the figure.

Describe the single transformation that would map 𝐴𝐡𝐢 onto 𝐴′𝐡′𝐢′.

  • AA 180∘ rotation about point 𝐷
  • BA 90∘ counterclockwise rotation about point 𝐷
  • CA 90∘ clockwise rotation about point 𝐷
  • DA 90∘ clockwise rotation about point 𝐸
  • EA 180∘ rotation about point 𝐸

Describe the single transformation that would map 𝐴′𝐡′𝐢′ onto 𝐴′′𝐡′′𝐢′′.

  • AA translation three left and three down
  • BA 90∘ clockwise rotation about point 𝐷
  • CA reflection in the line ⃖⃗𝐷𝐸
  • DA 90∘ counterclockwise rotation about point 𝐸
  • EA 90∘ clockwise rotation about point 𝐷

Hence, are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ congruent?

  • AYes
  • BNo

Q14:

The figure shows two triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐹.

Work out the measure of angle 𝐴𝐢𝐡.

Work out the measure of angle 𝐷𝐸𝐹.

What do you notice about the measures of the angles in both shapes?

  • AThe measures of the angles in both triangles depend on their lengths.
  • BThe measures of the angles in triangle 𝐴𝐡𝐢 are half the measures of the angles in triangle 𝐷𝐸𝐹.
  • CThe measures of the angles in triangle 𝐴𝐡𝐢 are double the measures of the angles in triangle 𝐷𝐸𝐹.
  • DThey are equal.

Are the two triangles similar?

  • ANo
  • BYes

Q15:

The figure shows three triangles: 𝐴𝐡𝐢, 𝐴𝐡𝐢, and 𝐴𝐡𝐢.

Are triangles 𝐴𝐡𝐢 and 𝐴𝐡𝐢 similar?

  • ANo
  • BYes

Justify your answer with one of the following reasons.

  • ATriangle 𝐴𝐡𝐢 can first be translated eight right and two down to 𝐴𝐡𝐢 and then 𝐴𝐡𝐢 can be reflected in the line ⃖⃗𝐸𝐹 onto 𝐴𝐡𝐢; hence, the triangles are similar.
  • BNo sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴𝐡𝐢 onto triangle 𝐴𝐡𝐢; therefore, the two triangles cannot be similar.

Q16:

The figure shows three triangles: 𝐴𝐡𝐢, 𝐴𝐡𝐢, and 𝐴𝐡𝐢.

Are triangles 𝐴𝐡𝐢 and 𝐴′′𝐡′′𝐢′′ similar?

  • AYes
  • BNo

Justify your answer with one of the following reasons.

  • ANo sequence of translations, reflections, rotations, or dilations exists that can map triangle 𝐴𝐡𝐢 onto triangle 𝐴′′𝐡′′𝐢′′; therefore, the two triangles cannot be similar.
  • BTriangle 𝐴𝐡𝐢 can first be translated eight right and two down to 𝐴′𝐡′𝐢′ and then 𝐴′𝐡′𝐢′ can be reflected in the line ⃖⃗𝐸𝐹 onto 𝐴′′𝐡′′𝐢′′; hence, the triangles are similar.

Q17:

A triangle 𝐴𝐡𝐢 has vertices at the points (βˆ’7,4), (βˆ’4,3), and (βˆ’1,3). A triangle 𝐷𝐸𝐹 has vertices at the points (1,βˆ’1), (4,βˆ’2), and (7,βˆ’2). By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • AThey are congruent.
  • BThey are not congruent.

Q18:

A triangle 𝐴𝐡𝐢 has vertices at the points (0,1), (1,3), and (βˆ’3,3). A triangle 𝐷𝐸𝐹 has vertices at the points (βˆ’2,βˆ’2), (βˆ’1,βˆ’4), and (βˆ’5,βˆ’4). By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • AThey are not congruent.
  • BThey are congruent.

Q19:

A triangle 𝐴𝐡𝐢 has vertices at the points (0,1), (1,2), and (5,2). A triangle 𝐷𝐸𝐹 has vertices at the points (0,βˆ’1), (1,βˆ’2), and (5,βˆ’1). By plotting the two triangles and using congruence transformations, decide if the two triangles are congruent.

  • AThey are congruent.
  • BThey are not congruent

Q20:

Triangle 𝐴𝐡𝐢 has been rotated to obtain triangle 𝐴𝐡𝐢 as seen in the given figure.

What is the length of 𝐡𝐢?

What is the length of 𝐴𝐢?

What type of triangle is 𝐴𝐡𝐢?

  • Aequilateral
  • Bscalene
  • Cisosceles

Q21:

In the given figure, △𝐷𝐸𝐢 is the image of △𝐴𝐡𝐢 by reflection in point 𝐢. Find the length of 𝐷𝐢, rounding your result to the nearest hundredth.

Q22:

Triangle 𝐴𝐡𝐢 is right-angled at 𝐡 with 𝐴𝐡=55cm and 𝐡𝐢=52cm. Let 𝑋 be the image of 𝐡 after a translation through 78 cm in the direction of 𝐡𝐴. Let π‘Œ be the image of 𝐡 under a rotation center 𝐴 through angle βˆ’90∘. Calculate the length π‘‹π‘Œ to the nearest hundredth.

Q23:

In the given figure, triangle 𝐴𝐡𝐢 has been reflected to triangle 𝐴′𝐡′𝐢′. The perimeter of triangle 𝐴𝐡𝐢 is 10.5. What is the perimeter of triangle 𝐴′𝐡′𝐢′?

Q24:

Triangle 𝐴𝐡𝐢 has been rotated about point 𝐷 to obtain triangle 𝐴′𝐡′𝐢′ as seen in the given figure. Work out the perimeter of triangle 𝐴𝐡𝐢.

Q25:

A quadrilateral 𝑄 has been dilated by a scale factor of 2 to 𝑄′. Are 𝑄 and 𝑄′ congruent?

  • ANo
  • BYes

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