Question

1. a ladder leans against a wall, forming a 90° angle with the ground. the angle between the ladder and the ground is 60°. which similarity criterion justifies that the triangle formed is similar to another triangle with the same angles?

options: a. asa (angle - side - angle), b. sss (side - side - side), c. sas (side - angle - side), d. aa (angle - angle)

1. rectangle e with side lengths 4 and 6 is dilated by a scale factor of 2. what are the side lengths of the resulting rectangle?

options: a. 8 and 12, b. 6 and 9, c. 10 and 15, d. 2 and 3

1. a 12 - inch tall model casts a shadow of 4 inches. a full - scale object casts a shadow of 16 feet. how tall is the full - scale object?

options: a. 48 feet, b. 64 feet, c. 36 feet, d. 45 feet

Explanation:

Question 13

The ladder, wall, and ground form a right triangle (90° with ground). The angle between ladder and ground is 60°, so the third angle (between ladder and wall) is 30° (since triangle angles sum to 180°: 180 - 90 - 60 = 30°). For triangle similarity, AA (Angle - Angle) criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Here, both triangles have a right angle (90°) and a 60° angle, so AA similarity applies. ASA, SSS, and SAS involve sides or specific angle - side configurations not met here.

Step 1: Recall dilation of a rectangle

When a rectangle is dilated by a scale factor, each side length is multiplied by the scale factor.

Step 2: Calculate new side lengths

The original side lengths are 4 and 6, and the scale factor is 2.
For the first side: $4\times2 = 8$
For the second side: $6\times2 = 12$

Step 1: Set up proportion

Let the height of the full - scale object be $h$ feet. Since the triangles formed by the object and its shadow are similar, we can set up a proportion: $\frac{\text{height of model}}{\text{shadow of model}}=\frac{\text{height of full - scale object}}{\text{shadow of full - scale object}}$.
The height of the model is 12 inches, the shadow of the model is 4 inches, and the shadow of the full - scale object is 16 feet. First, note that the units for the model are inches and for the full - scale object are feet, but since we are taking the ratio, the units of length (inches and feet) will cancel out as long as we are consistent. So we have $\frac{12}{4}=\frac{h}{16}$.

Step 2: Solve for $h$

Cross - multiply: $4h=12\times16$
$4h = 192$
Divide both sides by 4: $h=\frac{192}{4}=48$

Answer:

d. AA (Angle - Angle)

Question 14