Similar figures: corresponding sides, scale factor, and all circles are similar

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   Ari says △DEF is a scaled copy of △ABC. Which moves carry △ABC onto △DEF, and what do they show?

   • Slide 2 right and 1 up, then dilate by 2 about A: similar, with ratio 2
   • Dilate by 2 about A, then slide 2 right and 1 up: similar, with ratio 2
   • Dilate by 4 about A, then slide 2 right and 1 up: similar, with ratio 4
   • Dilate by 2 about A, then slide 2 right and 1 up: congruent, with equal sides
2. 2.

   Bo notices △DEF is turned over compared with △ABC. Which moves carry △ABC onto △DEF, and why are they similar?

   • Dilate by 2 about A, reflect across the x-axis, slide 2 right and 1 up; reflections keep ratios
   • Dilate by 2 about A, reflect across y = x, slide 2 right and 1 up; reflections keep ratios
   • Dilate by 2 about A, then slide 2 right and 1 up; no reflection is needed, since the sizes agree
   • Slide 2 right and 1 up, dilate by 2 about A, then reflect across y = x; reflections keep ratios
3. 3.

   Esme wants one dilation that carries △ABC onto △DEF. Is that possible?

   • Yes: dilate by 3 about A, since AC to DF is a ratio of 3
   • No: AB to DE is 2 but AC to DF is 3, so no single factor fits
   • Yes: dilate by 2 about A, because both are right triangles with a shared corner
   • Yes: dilate by 2.5, the average of 2 and 3
4. 4.

   Jun applies a dilation and then rigid motions to a triangle. Which statement is true for every such similarity transformation?

   • Areas are kept, because the shape is the same and just the position changes
   • Side lengths are kept, as in a rigid motion, so the image is congruent
   • Angle measures and ratios of sides are kept; lengths and areas are not
   • Perimeters are kept and angles are not, since a dilation bends the sides
5. 5.

   The sun makes the same angle with the ground for both. Both objects stand straight up. Kira stands a 1.5 m stick in the sun. Its shadow is 2 m. A tree's shadow is 14 m at the same time. Which setup finds the tree's height h?

   • Both are shadows, so the triangles are similar and 1.5/14 = 2/h
   • Shadows add to the height, so h = 14 − 2 + 1.5
   • Same angles give AA, but 1.5/2 = 14/h
   • Same sun angle and a right angle each give AA, so 1.5/2 = h/14
6. 6.

   Milo is 1.6 m tall and stands under a 4.8 m lamp post. Milo's shadow is 2 m. Which setup finds the distance d from the post's base to Milo?

   • The two right triangles share an angle (AA), so 4.8/(d + 2) = 1.6/2
   • The heights differ by 3.2, so d + 2 = 4.8 − 1.6 and the distance is 1.2 m
   • They share an angle, but 4.8/d = 1.6/2, taking the large base as d
   • They share an angle, but 4.8/(d + 2) = 2/1.6, with the sides in the other order
7. 7.

   The plan wrote 2/3 = h/30. Noah measured a 2 m pole's shadow, 3 m, at noon. A building's shadow was 30 m at 5 pm. Noah says the building is 20 m tall. What is wrong?

   • The shadows should be added to the heights first
   • The ratio should be 3/2 = h/30, with the shadow written over the height
   • The pole is shorter than the building, so it cannot be used
   • The sun angle was different, so the two right triangles are not similar
8. 8.

   similar, with scale factor 3 from the first to the second. Pia enlarges a poster to 3 times its width and height. Ink follows area, and the frame follows the edge length. How do the costs change?

   • Ink costs 3 times as much; the frame costs 3 times as much
   • Ink costs 9 times as much; the frame costs 3 times as much
   • Ink costs 9 times as much; the frame costs 9 times as much
   • Ink costs 27 times as much; the frame costs 9 times as much
9. 9.

   Ren makes a cube planter 30 cm on each edge, then one 60 cm on each edge. Soil follows volume, paint follows surface area, and tape follows edge length. How do they change?

   • Soil 4 times, paint 4 times, tape 2 times
   • Soil 8 times, paint 4 times, tape 2 times
   • Soil 8 times, paint 8 times, tape 4 times
   • Soil 2 times, paint 2 times, tape 2 times
10. 10.

    similar and solid, same material. Tala builds a solid 1:10 model statue that weighs 3 kg. The real statue is solid and made of the same stone. What does it weigh?

    • 30 kg, because each measurement grows 10 times and so does the weight
    • 9 kg, because the 3 kg model is multiplied by 3 for each of the three dimensions
    • 300 kg, because the surface grows 100 times and the weight follows it
    • 3000 kg, because weight follows volume: 3 × 10 × 10 × 10