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Dilations and Similarity: math test solutions, Grade 8 – download the PDF

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Test solutions Grade 8 : Dilations and Similarity — Zyro the alien explorer of Planète Maths

Test solutions with the detailed point scale. Add up your points and spot what to review.

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

1 Scale factor and lengths / 3 pts

  1. \(k = \dfrac{15}{6} = 2.5\). (1 pt)
  2. \(OB' = 2.5 \times 8 = 20\) cm. (1 pt)
  3. \(A'B' = 2.5 \times 4.2 = 10.5\) cm. (1 pt)

2 Dilation on the coordinate plane / 4 pts

  1. \(K(-2, 1) \to K'(-3, 1.5)\) (1 pt), \(L(4, 1) \to L'(6, 1.5)\) (1 pt), \(M(-2, -3) \to M'(-3, -4.5)\) (1 pt).
  2. \(KL = 4 - (-2) = 6\) and \(K'L' = 6 - (-3) = 9 = 1.5 \times 6\). (1 pt)

3 Similar triangles / 3 pts

  1. \(k = \dfrac{ST}{PQ} = \dfrac{6}{9} = \dfrac{2}{3}\). (1 pt)
  2. \(TU = \dfrac{2}{3} \times 12 = 8\) and \(SU = \dfrac{2}{3} \times 15 = 10\). (1 pt)
  3. Perimeters: \(9 + 12 + 15 = 36\) and \(6 + 8 + 10 = 24\), and \(\dfrac{24}{36} = \dfrac{2}{3}\): the perimeter is multiplied by the scale factor. (1 pt)

4 Angles of a triangle / 3 pts

  1. \(x + 2x + 10 + 3x + 20 = 180\), so \(6x = 150\) and \(x = 25\) (1 pt). The angles are \(25^\circ\), \(60^\circ\), and \(95^\circ\) (1 pt).
  2. The exterior angle is \(180 - 60 = 120^\circ\), which equals \(25 + 95\), the sum of the two remote interior angles. (1 pt)

5 Indirect measurement (calculator allowed) / 4 pts

  1. The person and the building are both vertical (right angles) and the sun’s rays make the same angle with the ground, so two angles match: the triangles are similar by AA. (1 pt)
  2. \(\dfrac{h}{36} = \dfrac{6}{4}\), so \(h = 36 \times 1.5 = 54\) ft. (2 pts)
  3. \(54 \times 0.3048 \approx 16.5\) m. (1 pt)

6 A sequence of transformations / 3 pts

  1. Translation: \((4, 2)\), \((6, 2)\), \((4, 5)\) (1 pt). Dilation: \(D'(8, 4)\), \(E'(12, 4)\), \(F'(8, 10)\) (1 pt).
  2. They are similar because the sequence includes a dilation. They are not congruent because the scale factor is \(2\), not \(1\): for example \(DE = 2\) but \(D'E' = 4\). (1 pt)
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