Congruent figures: corresponding parts and rigid motions
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
Noah claims any two congruent triangles are related by one flip. Does this pair support the claim?
- No: the sides of DEF are different from those of ABC
- No: DEF is ABC slid by ⟨5, 5⟩, and a flip would reverse the corner order
- Yes: a flip plus a slide works reliably, so one flip alone is surely enough
- Yes: congruent triangles are mirror images of each other, with the same size
Both triangles are right triangles with hypotenuse 13 and one leg 5. The other leg is not given. Two right triangles each have a hypotenuse of 13 and a leg of 5. Ari says they are congruent. Which reason is best?
- Two sides and a non-included angle (SSA) do fix a triangle
- The other leg is √(169 − 25) = 12 in both, so the sides match (SSS)
- Both have a right angle, so the angles match and AAA applies to the pair
- The hypotenuses match, so the two right triangles are congruent
Esme's map: a rigid motion takes A(1, 2) to A′(5, 2) and C(4, 1) to C′(2, 1), and leaves B(3, 5) where it is. What is it?
- A turn about B(3, 5), since B stays fixed
- A slide 4 units right
- A flip across the line y = 2
- A flip across the line x = 3
A rigid motion M₁ maps △ABC onto △DEF. A rigid motion M₂ maps △DEF onto △GHI. △ABC ≅ △DEF and △DEF ≅ △GHI. Bo says △ABC ≅ △GHI. Which justification is best?
- Congruence is equal area, and equal areas chain together from one triangle to the next
- The three triangles are named with letters in the same pattern, so they match
- It is true if △DEF is turned the same way as the others
- Doing M₁ then M₂ is one rigid motion taking △ABC to △GHI, so it keeps lengths and angles
Kira's tile WXYZ is a parallelogram with its corners in order. Diagonal WY cuts it into two triangles. Which rigid motion maps △WXY onto △YZW?
- A flip across the diagonal WY
- A slide along the diagonal WY
- A flip across the perpendicular bisector of WY
- A half turn (180°) about the midpoint of WY
Read this first
Statement
△ABC ≅ △DEF. AB = 2x + 4, BC = 3x − 1, AC = x + 9, in centimeters. Perimeter of △ABC: 36 cm.
Zee's tiles satisfy △ABC ≅ △DEF. AB = 2x + 4, BC = 3x − 1 and AC = x + 9. The perimeter of △ABC is 36 cm. How long is DF?
- 11 cm
- 12 cm
- 4 cm
- 13 cm
Pia claims a rigid motion never changes a segment's slope. A quarter turn takes the segment from (0, 0) to (2, 1) onto the segment from (0, 0) to (-1, 2). What does this show?
- The claim is false: the slope goes from 1/2 to -2, yet the length √5 stays
- The claim is true: both lengths are √5, so the slopes must match as well
- The claim is true: slope is a ratio, and a ratio does not change in any motion
- The claim is false: the length changes from √5 to 5
a slide. Step 2: a dilation with scale factor 1. Step 3: a rotation. Jun slides △ABC, enlarges the image with scale factor 1, then turns it. Is the final triangle congruent to △ABC?
- Yes: a scale factor of one changes nothing, so all three steps are rigid
- No: any dilation changes the size of a figure, so the result is bigger
- Yes: a dilation with any scale factor keeps the shape, so the result is congruent
- No: a dilation is not a slide, flip or turn, so the steps are not rigid
Nothing else is given about angles or the other diagonal. Milo's kite ABCD has AB = AD and CB = CD. Milo proves △ABC ≅ △ADC. Which proof is valid, and which motion shows it?
- SSS: AB = AD, CB = CD, AC shared; a flip across AC swaps B and D
- SAS: AB = AD, ∠B = ∠D and CB = CD, then a flip across AC
- ASA: AC bisects ∠A and ∠C, with AC shared between the triangles
- SSS: AB = AD, CB = CD and AC = BD, then a flip across AC
Ren counts the rigid motions that map an equilateral triangle exactly onto itself, including turning by 0°. How many are there?
- 3: just the three turns, since a flip changes the picture of the triangle
- 6: three turns (0°, 120°, 240°) and three flips
- 9: three turns, three flips and three slides along the sides
- 12: each of the 6 positions can be reached in two ways