Similar Triangles and Proportions: Geometry Math Test Answer Key – PDF Download

Question 1: Similarity by SAS (2 points)

1. \(\frac{DE}{AB}=\frac{12}{8}=\frac{3}{2}\) and \(\frac{DF}{AC}=\frac{18}{12}=\frac{3}{2}\), and the included angles are congruent (\(65^{\circ}\)). The triangles are similar by SAS.
2. ABC ~ DEF with scale factor \(\frac{3}{2}\).

Similar by SAS; ABC ~ DEF; k = 3/2.

Scoring: 1 point for both ratios and the criterion SAS; 1 point for a correctly ordered statement and k = 3/2.

Question 2: Similar by SSS (2 points)

1. \(\frac{12}{9}=\frac{16}{12}=\frac{20}{15}=\frac{4}{3}\): similar by SSS.
2. \(\frac{10}{5}=2,\ \frac{12}{6}=2,\ \frac{15}{7}\) ≈ \(2.14\): the ratios are not all equal, so not similar.

(a) similar, ratio 4/3; (b) not similar.

Scoring: 1 point for each correct decision supported by all three ratios.

Question 3: Triangle proportionality (3 points)

1. \(\frac{9}{6}=\frac{12}{y}\), so \(9y=72\) and \(y=8\).
2. \(AB=15\), so \(\frac{AD}{AB}=\frac{9}{15}=\frac{3}{5}=\frac{DE}{20}\) and \(DE=12\).
3. Angle A is shared and \(\angle ADE\) ≅ \(\angle ABC\) (corresponding angles), so AA.

EC = 8; DE = 12; AA.

Scoring: 1 point for EC = 8; 1 point for DE = 12 using the ratio of whole sides; 1 point for AA with a reason.

Question 4: Converse of the side-splitter theorem (2 points)

1. \(\frac{6}{10}=\frac{3}{5}\) and \(\frac{9}{15}=\frac{3}{5}\): parallel by the converse.
2. \(\frac{5}{7}\) ≈ \(0.714\) and \(\frac{8}{11}\) ≈ \(0.727\): not equal, so not parallel.

(a) parallel; (b) not parallel.

Scoring: 1 point for each correct conclusion with both ratios shown.

Question 5: Right triangle altitude (3 points)

1. \(CD^{2}=4\cdot 16=64\), so \(CD=8\).
2. \(AB=20\) and \(AC^{2}=AD\cdot AB=4\cdot 20=80\), so \(AC=\sqrt{80}=4\sqrt{5}\).
3. \(BC^{2}=DB\cdot AB=16\cdot 20=320\), so \(BC=\sqrt{320}=8\sqrt{5}\).

Check: \(80+320=400=20^{2}\). CD = 8, AC = \(4\sqrt{5}\), BC = \(8\sqrt{5}\).

Scoring: 1 point for each length, with the correct geometric mean setup.

Question 6: Complete a similarity proof (3 points)

Scoring: 1 point for each correct reason in steps 2, 3 and 4.

Question 7: Indirect measurement (2 points)

1. \(\frac{h}{36}=\frac{6}{8}\)
2. \(8h=216\), so \(h=27\).

The flagpole is 27 feet tall.

Scoring: 1 point for a correct proportion with matching quantities; 1 point for h = 27 ft.

Question 8: Similar triangles and algebra (2 points)

\(\frac{PQ}{ST}=\frac{QR}{TU}\): \(\frac{x+2}{10}=\frac{9}{15}\). Cross-multiply: \(15(x+2)=90\), so \(x+2=6\) and \(x=4\).

\(PQ=4+2=6\). x = 4 and PQ = 6.

Scoring: 1 point for the correct proportion; 1 point for x = 4 and PQ = 6.

Question 9: Geometric mean (1 point)

\(\sqrt{5\cdot 20}=\sqrt{100}=10\). 10

Scoring: 1 point for 10.

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