Congruent figures: corresponding parts and rigid motions

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   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 quarter turn about the origin
   • A flip across the line y = x
   • A half turn about the origin
2. 2.

   Noah claims any two congruent triangles are related by one flip. Does this pair support the claim?

   • 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
   • Yes: congruent triangles are mirror images of each other, with the same size
3. 3.

   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 hypotenuses match, so the two right triangles are congruent
   • 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
4. 4.

   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 flip across the line y = 2
   • A flip across the line x = 3
   • A slide 4 units right
   • A turn about B(3, 5), since B stays fixed
5. 5.

   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 diagonal WY
   • A flip across the perpendicular bisector of WY
   • A half turn (180°) about the midpoint of WY

6. Read this first

   Statement

   △ABC ≅ △DEF. AB = 2x + 4, BC = 3x − 1, AC = x + 9, in centimeters. Perimeter of △ABC: 36 cm.

7. 6.

   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?

   • 13 cm
   • 4 cm
   • 12 cm
   • 11 cm
8. 7.

   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: slope is a ratio, and a ratio does not change in any motion
   • The claim is false: the length changes from √5 to 5
   • The claim is true: both lengths are √5, so the slopes must match as well
9. 8.

   In isosceles △ABC with AB = AC, Tala draws the bisector AD of ∠A to the base BC and proves △ABD ≅ △ACD by SAS. Which facts does the proof need?

   • AB = AC (given), ∠ADB = ∠ADC (right angles), AD = AD (shared)
   • AB = AC (given), ∠BAD = ∠CAD (bisector), AD = AD (shared)
   • AB = AC (given), ∠B = ∠C (isosceles), AD = AD (shared)
   • AB = AC (given), BD = CD (midpoint), AD = AD (shared)
10. 9.

    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?

    • No: any dilation changes the size of a figure, so the result is bigger
    • Yes: a scale factor of one changes nothing, so all three steps are rigid
    • 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
11. 10.

    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
    • 3: just the three turns, since a flip changes the picture of the triangle
    • 9: three turns, three flips and three slides along the sides
    • 12: each of the 6 positions can be reached in two ways