Reflection and Congruence

14 Grade 10 Core Mathematics questions on Reflection and Congruence, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   3 marks

   A learner plots the corners of a school notice board as A(1, 2), B(4, 2) and C(4, 6) on a grid. State the coordinates of the images A', B' and C' when triangle ABC is reflected in the y-axis.

   (3 marks)

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2. Question 2

   1 mark

   On a plan of a tea-processing floor, a drying machine stands at M(5, 2). The plan is reflected in the line x = 1. State the coordinates of the image of M.

   A. (5, -2)

   B. (-5, 2)

   C. (-3, 2)

   D. (-4, 2)

   (1 mark)

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3. Question 3

   1 mark

   On a design for a sisal mat, the point A(2, 3) is mapped onto A'(3, 2) by a single reflection. State the equation of the mirror line.

   A. y = 0

   B. y = -x

   C. x = 0

   D. y = x

   (1 mark)

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4. Question 4

   2 marks

   Explain why a triangle and its image under a reflection are congruent, and yet the image cannot be made to fit exactly onto the object by sliding it across the paper without turning it over.

   (2 marks)

   Marking scheme

   Reflection keeps every side length and every angle size the same, so the object and image are congruent — 1 mark. Reflection reverses the sense (orientation): lettering that runs clockwise on the object runs anticlockwise on the image, so the image must be turned over to fit — 1 mark.

5. Question 5

   5 marks

   A timber roof truss contains triangle ABC in which AB = AC. D is the midpoint of BC and AD is joined. Show that triangle ABD is congruent to triangle ACD, state the congruence test used, and hence state one conclusion about angle ADB.

   (5 marks)

   Marking scheme

   AB = AC (given, the truss is isosceles) — 1 mark. BD = DC (D is the midpoint of BC) — 1 mark. AD is common to both triangles — 1 mark. Therefore triangle ABD is congruent to triangle ACD by SSS — 1 mark. Hence angle ADB = angle ADC; as adjacent angles on the straight line BC they add up to 180°, so each is 90° and AD is perpendicular to BC — 1 mark.

6. Question 6

   1 mark

   A bead on a Maasai beadwork pattern is at the point B(2, 5) on a grid. Identify the coordinates of the image of B after a reflection in the line y = -x.

   A. (5, -2)

   B. (-2, -5)

   C. (5, 2)

   D. (-5, -2)

   (1 mark)

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7. Question 7

   4 marks

   A fence post stands at P(7, -2) on a grid plan of a maize plot. The plan is reflected in the line x = 3.

   (a) State the coordinates of the image P'.

   (b) State the distance of P from the mirror line.

   (c) State the coordinates of the midpoint of PP'.

   (d) State the relationship between the line x = 3 and the line PP'.

   (4 marks)

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8. Question 8

   4 marks

   A logo printed on a tea packet has a point A(2, 1) whose image under a single reflection is A'(2, 7). Determine the equation of the mirror line and explain how you obtained it.

   (4 marks)

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9. Question 9

   3 marks

   Outline three different sets of conditions, each using three facts, that are sufficient to show that two triangles are congruent.

   (3 marks)

   Marking scheme

   Award 1 mark for each correctly stated test, maximum 3. SSS — three pairs of corresponding sides equal. SAS — two pairs of corresponding sides equal and the angle between them equal. ASA (or AAS) — two pairs of corresponding angles equal and one pair of corresponding sides equal. RHS — in right-angled triangles, the hypotenuse and one other side equal. Do not credit AAA.

10. Question 10

    1 mark

    Identify the set of conditions that is NOT sufficient to prove that two triangles are congruent.

    A. Three pairs of corresponding angles equal

    B. Three pairs of corresponding sides equal

    C. Two sides and the included angle equal

    D. A right angle, the hypotenuse and one other side equal

    (1 mark)

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11. Question 11

    2 marks

    Distinguish between congruent figures and similar figures.

    (2 marks)

    Marking scheme

    Congruent figures have the same shape AND the same size: corresponding sides are equal and corresponding angles are equal — 1 mark. Similar figures have the same shape but not necessarily the same size: corresponding angles are equal while corresponding sides are in a constant ratio (scale factor) — 1 mark.

12. Question 12

    1 mark

    A triangle drawn on a grid plan of a school compound is reflected in the line x = 3. One of its vertices maps onto itself. Identify that vertex.

    A. (5, 3)

    B. (-3, 5)

    C. (3, 5)

    D. (0, 3)

    (1 mark)

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13. Question 13

    1 mark

    A surveyor plots a borehole at the point P(-4, 6) on a grid map of a farm in Kitui. Identify the coordinates of the image of P after a reflection in the x-axis.

    A. (6, -4)

    B. (4, 6)

    C. (4, -6)

    D. (-4, -6)

    (1 mark)

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14. Question 14

    1 mark

    In triangles ABC and PQR, AB = PQ, BC = QR and angle ABC = angle PQR. Identify the congruence test that proves the two triangles congruent.

    A. SSS

    B. SAS

    C. RHS

    D. ASA

    (1 mark)

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