Testinar Grade 8 math

Similarity and Dilations

Learn to reason about and solve Grade 8 problems involving Similarity and Dilations.

CH05-04 Geometry and Measurement

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Date
Score: ____ / 8

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Two triangles are shown below. Triangle \(ABC\) has sides \(3, 4, 5\). Triangle \(DEF\) has sides \(6, 8, 10\). Are these triangles similar?
1. No, because the sides are different lengths
2. Yes, each side of \(DEF\) is twice the corresponding side of \(ABC\)
3. No, the angles are not equal
4. Cannot be determined
Two similar triangles are shown. Triangle \(ABC\) has sides \(4 \text{ cm}\) and \(5 \text{ cm}\). Triangle \(DEF\) has a corresponding side of \(8 \text{ cm}\). Find the length of the side in \(DEF\) corresponding to the \(5 \text{ cm}\) side in \(ABC\).
1. \(10 \text{ cm}\)
2. \(6.5 \text{ cm}\)
3. \(9 \text{ cm}\)
4. \(2.5 \text{ cm}\)
Figure \(P\) has area \(16 \text{ m}^2\) and Figure \(Q\) (similar to \(P\)) has area \(64 \text{ m}^2\). What is the scale factor from \(P\) to \(Q\)?
1. \(2\)
2. \(4\)
3. \(8\)
4. \(16\)
Two triangles both have angles measuring 45°, 60°, and 75°. Are these triangles necessarily similar?
1. No, the sides must also be proportional
2. No, only two angles are given
3. Yes, angles alone determine similarity (AAA)
4. Only if the triangles are congruent
Point \(A\) is at \((2, 3)\) and is dilated by a scale factor of \(2\) from the origin. Where is the dilated point \(A'\)?
1. \((1, 1.5)\)
2. \((4, 6)\)
3. \((2, 3)\)
4. \((2, 5)\)
Two rectangles are shown with proportional dimensions. Which similarity statement is correct?
1. Rectangle 1 \(\sim\) Rectangle 2 with scale factor \(1.5\)
2. The rectangles are not similar
3. Rectangle 1 \(\sim\) Rectangle 2 with scale factor \(0.5\)
4. They are congruent
Two quadrilaterals are shown. Quadrilateral \(ABCD\) has sides \(3, 4, 5, 6\). Quadrilateral \(EFGH\) has corresponding sides \(6, 8, 10, 12\), and their corresponding angles are congruent. Are they similar?
1. No, angles are different
2. Yes, all sides are in ratio \(1:2\)
3. Cannot tell without angles
4. Only if both are trapezoids
A dilated figure has a side length of \(12 \text{ cm}\) and was created with a scale factor of \(1.5\) from the original. What was the original side length?
1. \(8 \text{ cm}\)
2. \(18 \text{ cm}\)
3. \(6 \text{ cm}\)
4. \(13.5 \text{ cm}\)

Teacher and parent reference

Answer key and explanations

B. Yes, each side of \(DEF\) is twice the corresponding side of \(ABC\)
Compute ratios of corresponding sides: \(\frac{DE}{AB} = \frac{6}{3} = 2\), \(\frac{DF}{AC} = \frac{8}{4} = 2\), \(\frac{EF}{BC} = \frac{10}{5} = 2\). All ratios are constant (= 2), confirming similarity by SSS Similarity. Angles are also congruent in similar figures.
A. \(10 \text{ cm}\)
Scale factor from \(ABC\) to \(DEF\) is \(\frac{8}{4} = 2\). The corresponding side is \(5 \times 2 = 10 \text{ cm}\).
A. \(2\)
If area ratio is \(\frac{64}{16} = 4\), then the linear scale factor is \(\sqrt{4} = 2\).
C. Yes, angles alone determine similarity (AAA)
If two triangles have all three corresponding angles congruent, they are similar by AAA (Angle-Angle-Angle). Here, both triangles have the same three angles, so they must be similar regardless of side lengths.
B. \((4, 6)\)
Multiply each coordinate by the scale factor: \((2 \times 2, 3 \times 2) = (4, 6)\).
A. Rectangle 1 \(\sim\) Rectangle 2 with scale factor \(1.5\)
Ratio of corresponding sides: \(\frac{9}{6} = 1.5\) and \(\frac{6}{4} = 1.5\). Constant ratio confirms similarity.
B. Yes, all sides are in ratio \(1:2\)
Verify each side ratio: \(\frac{6}{3} = 2\), \(\frac{8}{4} = 2\), \(\frac{10}{5} = 2\), \(\frac{12}{6} = 2\). All ratios are constant, so sides are proportional (scale factor 2). Combined with congruent corresponding angles, the quadrilaterals are similar.
A. \(8 \text{ cm}\)
If the new length is \(12\) and the scale factor is \(1.5\), then original \(= \frac{12}{1.5} = 8 \text{ cm}\). Distractor D adds the scale factor instead of dividing.