Math: Similar and Congruent Figures
Comparing shape, size, and scale
Math: Similar and Congruent Figures
Comparing shape, size, and scale
Math - Grade 6-8
- 1
Triangle A has side lengths 3 cm, 4 cm, and 5 cm. Triangle B has side lengths 6 cm, 8 cm, and 10 cm. Are the triangles similar, congruent, or neither? Explain.
Compare each side in one triangle to the matching side in the other triangle.
The triangles are similar. Each side length in Triangle B is 2 times the matching side length in Triangle A, so they have the same shape but different sizes. - 2
A square has side length 7 cm. Another square has side length 7 cm. Are the squares similar, congruent, or both? Explain.
The squares are both similar and congruent. All squares are similar because they have the same angle measures and shape, and these two are congruent because their corresponding sides are equal. - 3
Rectangle P is 3 units wide and 8 units long. Rectangle Q is 6 units wide and 16 units long. Are the rectangles similar, congruent, or neither?
Check whether both dimensions are multiplied by the same number.
The rectangles are similar. The width and length of Rectangle Q are both 2 times the width and length of Rectangle P, so the side lengths are proportional. - 4
Rectangle M is 4 units by 9 units. Rectangle N is 8 units by 15 units. Are the rectangles similar, congruent, or neither? Explain.
The rectangles are neither similar nor congruent. The side lengths are not equal, so they are not congruent, and the ratios 8/4 and 15/9 are not the same, so they are not similar. - 5
Two triangles are congruent. One triangle has side lengths 5 cm, 12 cm, and 13 cm. What are the side lengths of the other triangle?
Congruent figures match exactly.
The other triangle also has side lengths 5 cm, 12 cm, and 13 cm. Congruent figures have the same size and shape, so all corresponding sides are equal. - 6
A photo that is 4 inches wide and 6 inches tall is enlarged so that the new width is 10 inches. What is the scale factor? What is the new height?
The scale factor is 2.5 because 10 divided by 4 equals 2.5. The new height is 15 inches because 6 times 2.5 equals 15. - 7
Figure X and Figure Y are similar. A side in Figure X is 9 cm, and the matching side in Figure Y is 12 cm. What is the scale factor from Figure X to Figure Y?
Scale factor equals new length divided by original length.
The scale factor from Figure X to Figure Y is 4/3, or about 1.33. This is found by dividing 12 by 9. - 8
A triangle is enlarged by a scale factor of 3. If one side of the original triangle is 7 units, what is the length of the matching side in the enlarged triangle?
The matching side in the enlarged triangle is 21 units. Multiplying 7 by the scale factor 3 gives 21. - 9
A rectangle is reduced by a scale factor of 1/2. If the original length is 18 cm and the original width is 10 cm, what are the new dimensions?
A scale factor less than 1 makes the figure smaller.
The new length is 9 cm and the new width is 5 cm. Each original dimension is multiplied by 1/2. - 10
Explain the difference between similar figures and congruent figures.
Similar figures have the same shape, but they may have different sizes. Congruent figures have the same shape and the same size, so all corresponding sides and angles match exactly. - 11
Two pentagons have equal corresponding angles, and each side length in the larger pentagon is 3 times the matching side length in the smaller pentagon. Are the pentagons similar, congruent, or neither?
Similar figures need equal corresponding angles and proportional side lengths.
The pentagons are similar. Their corresponding angles are equal and their corresponding side lengths have a constant ratio of 3, so they have the same shape but different sizes. - 12
Shape A has side lengths 2 cm, 2 cm, 3 cm, and 3 cm. Shape B has side lengths 4 cm, 4 cm, 6 cm, and 6 cm. If the corresponding angles are equal, are the shapes similar, congruent, or both?
The shapes are similar. Each side length in Shape B is 2 times the matching side length in Shape A, so the sides are proportional, but the figures are not congruent because the sizes are different. - 13
A map uses a scale where 1 inch represents 5 miles. If two towns are 3.5 inches apart on the map, how far apart are they in real life?
Use the map scale as a multiplication rule.
The towns are 17.5 miles apart in real life. Multiplying 3.5 inches by 5 miles per inch gives 17.5 miles. - 14
A model car is built at a scale of 1:20. If the real car is 180 inches long, how long is the model car?
The model car is 9 inches long. A scale of 1:20 means the model length is the real length divided by 20, and 180 divided by 20 equals 9. - 15
Two figures have the same shape, but one figure is twice as large as the other. Can the figures be congruent? Can they be similar? Explain.
Think about which word requires an exact match in size.
The figures cannot be congruent because congruent figures must be the same size. The figures can be similar because similar figures can have the same shape with different sizes.