Question

1. a tree has a shadow 11 meters long and the height of the tree is unknown. a pole that is 5 meters tall casts a shadow 6 meters long. are the triangles formed by the tree and the pole similar?

a. no, the sides are not proportional
b. yes, by the aa criterion
c. yes, by the sas criterion
d. no, the angles are not congruent

1. a triangular billboard frame has two sides measuring 10 meters and 12 meters, with the included angle measuring 60°. another frame has sides of 8 meters and 9 meters with the same 60° included angle.

a. yes, by the sss criterion
b. no, the sides are not proportional
c. yes, by the sas criterion
d. no, the angles do not match

1. two triangles are similar, and one triangle has sides 8 cm, 12 cm, and 16 cm. the corresponding sides of the other triangle are proportional, with the shortest side measuring 6 cm. what is the length of the longest side?

a. 12 cm
b. 20 cm
c. 24 cm
d. 18 cm

1. a flagpole has sides that form a triangle with the ground. the height of the flagpole is 8 meters, and the distance to the tip of its shadow is 10 meters, forming a 60° angle with the ground. a nearby tree has a height of 12 meters and a shadow of 15 meters. are the triangles similar?

a. no, the angles are not congruent
b. yes, by the sss criterion
c. no, the sides are not proportional
d. yes, by the sas criterion

1. when proving similarity using the aa criterion, how many angle pairs need to be equal?

a. all corresponding angles must be proportional
b. 2
c. 3
d. 1

1. which of the following best describes the triangle proportionality theorem?

a. the sum of the interior angles of a triangle is 180 degrees.
b. if two triangles are similar, their corresponding angles are equal.
c. if two triangles are congruent, their corresponding sides are proportional.
d. a line parallel to one side of a triangle divides the other two sides proportionally.

1. if two triangles are similar, what must be true about their corresponding angles?

a. they are complementary.
b. they are equal.
c. they are supplementary.
d. they are congruent.

Explanation:

Step1: Find the scale factor

The first triangle has sides \(8\) cm, \(12\) cm, and \(16\) cm. The shortest side of the second triangle is \(6\) cm. The scale factor \(k\) is \(\frac{6}{8}=\frac{3}{4}\).

Step2: Calculate the length of the corresponding side

We want to find the length of the side corresponding to \(16\) cm. Using the scale - factor formula \(y = kx\) (where \(x\) is the side of the first triangle and \(y\) is the side of the second triangle), substitute \(x = 16\) and \(k=\frac{3}{4}\). Then \(y=\frac{3}{4}\times16\).

$$y=\frac{3\times16}{4}=3\times4 = 12$$

Answer:

a. \(12\) cm