5.3 Academic Practice With Similarity

Explanation
Equilateral triangles are always similar because they have three equal sides and three equal angles. Similarity in triangles means that their corresponding angles are equal and their corresponding sides are proportional. Since all sides and angles of an equilateral triangle are equal, any two equilateral triangles will have the same shape and size, making them similar. On the other hand, right triangles and isosceles triangles can vary in shape and size, so they are not always similar.

Explanation
In a similar triangle, the corresponding sides are proportional. To find the length of the longest side of the similar triangle, we can set up a proportion using the given information. Let x be the length of the longest side of the similar triangle. Then, we can set up the proportion 5/15 = x/10. 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. They have the same shape, but their sizes may be different. The corresponding angles of similar triangles are equal, but the lengths of their corresponding sides are proportional. This means that if we scale one triangle up or down, the other triangle will still be similar, but not identical in size. Therefore, the statement that similar triangles are exactly the same shape and size is false.

Explanation
In the given diagram, DE is parallel to AC. We are given that BD = 4, DA = 6, and EC = 8. To find BC, we can use the similar triangles formed by DEB and CEB. Since DE is parallel to AC, we can use the property of corresponding angles to determine that angle DEB is equal to angle CEB. Using the property of similar triangles, we can set up the following proportion: DE/CE = BD/BC. Plugging in the given values, we have 4/8 = 6/BC. Cross multiplying and solving for BC, we get BC = (8*6)/4 = 12. This value rounded to the nearest tenth is 12.0, which is closest to 5.3.

Explanation
In the given diagram, DE is parallel to AC, which means that triangle BDE is similar to triangle BAC by the AA similarity theorem. Since BD = 4 and DA = 8, we can use the proportional sides of similar triangles to find the length of BC. The ratio of the corresponding sides of triangle BDE to triangle BAC is 4/8 = 1/2. Therefore, the length of BC is half the length of AC. Since EC = 9, AC = 9 + 9 = 18. Half of 18 is 9, so BC is equal to 9. Therefore, the correct answer is 13.5.

Explanation
The two ladders are leaned against the wall at the same angle with the ground, which means they form similar triangles. The ratio of the length of the ladder to the height it reaches on the wall is the same for both ladders. In this case, the ratio is 10/8. To find how much further up the wall the 18' ladder reaches, we can set up a proportion: (10/8) = (18/x). Solving for x, we find that x is equal to 14.4 ft. Therefore, the 18' ladder reaches 14.4 ft further up the wall.

Explanation
The height of the tree can be determined using a proportion. Since the shadow of the boy is 8' and his height is 5', we can set up the proportion: (Height of tree)/(Shadow of tree) = (Height of boy)/(Shadow of boy). Plugging in the given values, we have (Height of tree)/28 = 5/8. Solving for the height of the tree, we get Height of tree = (28 * 5)/8 = 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, we divide the length of the larger polygon's side by the length of the smaller polygon's side: 9/6 = 3/2. Therefore, the ratio of similitude between the larger and smaller polygons is 3:2.

Explanation
The triangles are similar because they both have a 90-degree angle and a 35-degree angle. The third angle in triangle ABC would be 55 degrees (180 - 90 - 35), and the third angle in triangle DEF would be 90 degrees (180 - 35 - 55). Since the corresponding angles in both triangles are equal, they are similar.

Explanation
Since angle A and angle A' are each 59°, we can conclude that triangle ABC and triangle A'BC are congruent. This means that side AC is equal to side A'C. Therefore, AC is equal to 12.

Explanation
The answer is "yes" because triangle ABC and triangle ADE are similar if their corresponding angles are equal and their corresponding sides are in proportion.

Explanation
The SSS method can be used to prove that these triangles are similar. SSS stands for Side-Side-Side, which means that if the corresponding sides of two triangles are proportional, then the triangles are similar. In this case, if all three sides of the triangles are proportional, then they can be proven to be similar using the SSS method.

Explanation
The answer SAS stands for "Side-Angle-Side" and is a method used to prove that two triangles are similar. This method states that if two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar. Therefore, by using the SAS method, it would be possible to prove the given triangles similar if the conditions are met.

Explanation
The answer "AA" refers to the Angle-Angle similarity criterion. This criterion states 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 corresponding angles of the triangles are congruent, it would be possible to prove the triangles similar using the AA criterion.

Explanation
In a triangle with sides in the ratio 3:4:5, we can assign variables to the measures of the sides. Let the shortest side be 3x. Then, the other two sides would be 4x and 5x. The perimeter of the triangle is the sum of all three sides, which is 3x + 4x + 5x = 12x. We are given that the perimeter is 48, so we can solve for x by setting up the equation 12x = 48. Solving for x, we find x = 4. Therefore, the measure of the shortest side is 3x = 3(4) = 12.

Explanation
The correct answer is A and C only. This means that both options A and C are equivalent proportions. The question does not provide the original proportion, so we cannot determine the exact values. However, we can conclude that options B and none of the above are not equivalent proportions.