Congruent figures: corresponding parts and rigid motions
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
Luz reflects P(3, -2) across the x-axis, then across the y-axis. Which single rigid motion does the pair equal?
- A flip across the y-axis alone, leaving the y values
- A half turn about the origin
- A quarter turn about the origin
- A flip across the line y = x
Noah claims any two congruent triangles are related by one flip. Does this pair support the claim?
- Yes: congruent triangles are mirror images of each other, with the same size
- 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
- No: the sides of DEF are different from those of ABC
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?
- The other leg is √(169 − 25) = 12 in both, so the sides match (SSS)
- Two sides and a non-included angle (SSA) do fix a triangle
- 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 slide 4 units right
- A flip across the line y = 2
- A turn about B(3, 5), since B stays fixed
- 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?
- 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
- 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
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 slide along the diagonal WY
- A flip across the perpendicular bisector of WY
- A half turn (180°) about the midpoint of WY
- A flip across the diagonal 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
- 13 cm
- 12 cm
- 4 cm
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 dilation with any scale factor keeps the shape, so the result is congruent
- 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
- 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?
- ASA: AC bisects ∠A and ∠C, with AC shared between the triangles
- 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
- 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?
- 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
- 3: just the three turns, since a flip changes the picture of the triangle