5.3 Preap Practice With Similarity

Explanation
Equilateral triangles are always similar because they have equal angles and equal sides. The ratio of the lengths of corresponding sides in any two equilateral triangles will always be the same, making them similar. Isosceles triangles can be similar if they have equal angles, but they do not have equal sides. Right triangles can be similar if they have equal angles, but they do not have equal sides unless they are also isosceles. Therefore, equilateral triangles are the only triangles that are always similar.

Explanation
The sides of a triangle are in proportion to each other in similar triangles. In this case, the ratio of the sides of the original triangle is 5:6:10. To find the longest side of the similar triangle, we need to find the corresponding side in the original triangle. Since the shortest side of the similar triangle is 15, we can set up a proportion: 5/15 = 10/x. Solving for x, we get x = 30. Therefore, the length of the longest side of the similar triangle is 30.

Explanation
Similar triangles are not exactly the same shape and size. Similar triangles have the same shape, meaning that their corresponding angles are equal, but their sizes can be different. The sides of similar triangles are proportional to each other, meaning that the ratio of the lengths of corresponding sides is the same for all pairs of corresponding sides. Therefore, similar triangles can be larger or smaller than each other, but they will have the same shape.

Explanation
In the given diagram, DE is parallel to AB, which means that triangle ABD and triangle CDE are similar. Using the property of similar triangles, we can set up the proportion BD/DA = EC/BC. Substituting the given values, we have 4/6 = 8/BC. Cross-multiplying and solving for BC, we get BC = 12. Plugging this value into the answer choices, we find that the closest value is 5.3.

Explanation
In the given diagram, DE is parallel to AC, which means that triangle ADE is similar to triangle ABC. Using the property of similar triangles, we can set up the following proportion: AB/AD = BC/DE. Plugging in the given values, we get AB/8 = BC/9. Cross multiplying and solving for BC, we find that BC = (9/8) * AB. Since AB = BD + DA = 4 + 8 = 12, we can substitute this value into the equation to find BC = (9/8) * 12 = 13.5. Therefore, the correct answer is 13.5.

Explanation
The two ladders are leaned against the wall at the same angle with the ground. This means that the ratio of the length of the ladder to the height it reaches on the wall is the same for both ladders.

Given that the 10' ladder reaches 8' up the wall, we can set up a proportion to find the height reached by the 18' ladder.

10/8 = 18/x

Solving for x, we find that the 18' ladder reaches 14.4' up the wall. Therefore, the answer is 6.4 ft.

Explanation
The height of the tree can be determined using the concept of similar triangles. The ratio of the length of the boy's shadow to his height is the same as the ratio of the length of the tree's shadow to its height. So, we can set up the proportion: (8/5) = (28/x), where x is the height of the tree. Solving for x, we get x = 17.5. Therefore, the height of the tree is 17.5 feet.

Explanation
The ratio of similitude between two similar polygons is determined by comparing the lengths of their corresponding sides. In this case, the larger polygon has a side length of 9 inches, while the smaller polygon has a side length of 6 inches. To find the ratio of similitude, we divide the length of the larger polygon's side by the length of the smaller polygon's side. Thus, the ratio of similitude is 9/6, which simplifies to 3/2. Therefore, the correct answer is 3:2.

Explanation
The triangles ABC and DEF are similar because they both have a pair of congruent angles. Angle B in triangle ABC is congruent to angle E in triangle DEF, and angle A in triangle ABC is congruent to angle F in triangle DEF. This satisfies the Angle-Angle (AA) similarity criterion, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Therefore, the triangles ABC and DEF are similar.

Explanation
Since angle A and angle A' are each 59°, it means that angle A and angle A' are congruent. In a triangle, the sum of all angles is 180°. Therefore, angle C is 180° - 59° - 59° = 62°. AC is the side opposite to angle C in the triangle. To find AC, we can use the law of sines, which states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides in a triangle. Using the law of sines, we can set up the following equation: AC/sin(62°) = 10/sin(59°). Solving for AC, we get AC ≈ 12.

Explanation
Triangle ABC is similar to triangle ADE because they have corresponding angles that are equal and corresponding sides that are proportional. This means that the two triangles have the same shape, but possibly different sizes.

Explanation
The answer (6,3) is the only point that is different from the other two points in the question. The first two points have the same y-coordinate of 6, while the third point has a different y-coordinate of 3. Therefore, the correct answer is (6,3).

Explanation
The answer is SSS (Side-Side-Side) because if all three sides of two triangles are proportional to each other, then the triangles are similar. This means that the corresponding angles will also be equal.

Explanation
The correct answer is SAS (Side-Angle-Side). This method can be used to prove that two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent.

Explanation
The correct answer is AA. AA stands for Angle-Angle, which means that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. In this case, if the angles of the two triangles are congruent, then the triangles can be proven to be similar using the AA method.

Explanation
The scale factor used to reduce the poster is 2:15. This means that for every 2 units on the original poster, there will be 15 units on the flyer. In this case, the dimensions of the poster are reduced by a factor of 2 in both length and width to create the flyer.