A tree in a schoolyard casts a 60-inch shadow at the same time a nearby student casts a 17-inch shadow. If the student is 68 inches tall, how many feet tall is the tree?

Answer 1

To find the height of the tree, we can set up a proportion based on the similar triangles formed by the heights of the objects and their corresponding shadow lengths.

Let \( h \) be the height of the tree in inches. According to the problem, we can set up the proportion as follows:

\[ \frac{\text{Height of the tree}}{\text{Length of the tree's shadow}} = \frac{\text{Height of the student}}{\text{Length of the student's shadow}} \]

Substituting the known values into the equation, we have:

\[ \frac{h}{60} = \frac{68}{17} \]

Now we can simplify the right side:

\[ \frac{68}{17} = 4 \]

Now we can rewrite the proportion:

\[ \frac{h}{60} = 4 \]

To find \( h \), we can multiply both sides by 60:

\[ h = 4 \times 60 \]

Calculating this gives:

\[ h = 240 \]

So, the height of the tree is 240 inches. To convert this height into feet, since there are 12 inches in a foot, we divide by 12:

\[ \frac{240}{12} = 20 \]

Thus, the height of the tree in feet is

\[ \boxed{20} \]

Answer 2

If one image can be obtained from another by a sequence of rotations, reflections, or translations the images are

Answer 3

If one image can be obtained from another by a sequence of rotations, reflections, or translations, the images are said to be congruent. Congruent figures are identical in shape and size, and their corresponding parts (sides and angles) are equal. The transformations mentioned—rotations, reflections, and translations—are all isometric transformations, meaning they preserve distances and angles, thus maintaining congruence.

Answer 4

states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Answer 5

The statement you're referring to is known as the Angle-Angle (AA) similarity postulate. This postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

This means that their corresponding sides are in proportion, and the triangles have the same shape but may differ in size. The AA similarity criteria can be used to prove that triangles are similar without needing to know the lengths of their sides.