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

1 Scale factor and lengths / 3 pts

A dilation has center \(O\). Point \(A\) is \(6\) cm from \(O\) and its image \(A'\) is \(15\) cm from \(O\).

1. Find the scale factor \(k\).
2. Point \(B\) is \(8\) cm from \(O\). How far is \(B'\) from \(O\)?
3. \(AB = 4.2\) cm. Find \(A'B'\).

2 Dilation on the coordinate plane / 4 pts

Triangle \(KLM\) is shown below. It is dilated with center the origin and scale factor \(1.5\).

1. Find the coordinates of \(K'\), \(L'\), and \(M'\).
2. Compare \(KL\) and \(K'L'\).

3 Similar triangles / 3 pts

\(\triangle PQR \sim \triangle STU\) with \(PQ = 9\), \(QR = 12\), \(PR = 15\), and \(ST = 6\).

1. Find the scale factor from \(PQR\) to \(STU\).
2. Find \(TU\) and \(SU\).
3. Compare the perimeters of the two triangles.

4 Angles of a triangle / 3 pts

A triangle has angles \(x^\circ\), \((2x + 10)^\circ\), and \((3x + 20)^\circ\).

1. Find \(x\) and the three angles.
2. Extend a side to create the exterior angle at the vertex of the \(60^\circ\) angle (the extended side touches that vertex). Find this exterior angle and explain how it relates to the other two angles.

5 Indirect measurement (calculator allowed) / 4 pts

At the same time of day, a \(6\)-ft-tall person casts a \(4\)-ft shadow and a building casts a \(36\)-ft shadow.

1. Explain why the triangles formed are similar.
2. Find the height of the building in feet.
3. Convert it to meters (\(1\) ft \(= 0.3048\) m) and round to the nearest tenth.

6 A sequence of transformations / 3 pts

Triangle \(DEF\) has vertices \(D(1, 2)\), \(E(3, 2)\), \(F(1, 5)\). It is translated \(3\) units to the right, then dilated with center the origin and scale factor \(2\).

1. Find the vertices of the final image \(D'E'F'\).
2. Are \(DEF\) and \(D'E'F'\) similar? Congruent? Explain.