Similarity and Dilations: math test solutions, Grade 10 – download the PDF

1 Dilation of a triangle / 3 pts

\((x,y)\mapsto(1.5x,1.5y)\).

\(A'=(6,9)\) (1 pt), \(B'=(-3,12)\) (1 pt), \(C'=(15,-9)\) (1 pt).

2 Are they similar? / 4 pts

1. \(\dfrac{12}{8}=\dfrac{18}{12}=\dfrac{21}{14}=1.5\). The three ratios are equal, so the triangles are similar by SSS (2 pts).
2. Perimeter of \(PQR\): \(8+12+14=34\). The perimeter ratio is \(k=1.5\), so the perimeter of \(STU\) is \(34\times1.5=51\) (2 pts). Check: \(12+18+21=51\).

3 Parallel segment in a triangle / 4 pts

1. \(AB=15\) and \(\dfrac{AD}{AB}=\dfrac{DE}{BC}\), so \(\dfrac{9}{15}=\dfrac{12}{BC}\) and \(BC=\dfrac{12\cdot15}{9}=20\) (2 pts).
2. \(\dfrac{AD}{DB}=\dfrac{AE}{EC}\), so \(\dfrac96=\dfrac{AE}{4}\) and \(AE=6\) (2 pts).

4 Angle bisector / 3 pts

1. \(\dfrac{BD}{DC}=\dfrac{AB}{AC}=\dfrac{14}{21}=\dfrac23\) (1 pt).
2. \(BD=\dfrac25\cdot20=8\) and \(DC=\dfrac35\cdot20=12\) (2 pts). Check: \(8+12=20\).

5 Triangular lots / 3 pts

1. \(k=\dfrac{45}{30}=1.5\) (1 pt).
2. Area: \(600\times1.5^2=600\times2.25=1350\) ft² (1 pt).
3. Perimeter: \(100\times1.5=150\) ft (1 pt).

6 A bow-tie proof / 3 pts

1. \(\angle CAE\cong\angle DBE\) (alternate interior angles, \(AC\parallel BD\)) and \(\angle AEC\cong\angle BED\) (vertical angles). By AA, \(\triangle EAC\sim\triangle EBD\) (2 pts).
2. \(\dfrac{EA}{EB}=\dfrac{EC}{ED}\), so \(\dfrac{10}{15}=\dfrac{8}{ED}\) and \(ED=12\) (1 pt).