Question

a building casts a shadow 20 meters long, and a 4 - meter pole casts a shadow 4 meters long. what is the height of the building?
a. 20 meters
b. 15 meters
c. 16 meters
d. 10 meters

in $\triangle abc \sim \triangle def$, $ab = 10$, $ac = 15$, and $de = 5$. what is the length of $df$?
a. 20
b. 10
c. 7.5
d. 12

a flagpole casts a shadow 24 meters long, and a nearby pole 3 meters tall casts a shadow 4 meters long. which criterion confirms the similarity of the triangles?
a. sas criterion
b. sss criterion
c. right triangle criterion
d. aa criterion

Explanation:

First Question (Building and Pole Shadow)

Step1: Set up proportion

Let \( h \) be the height of the building. The ratio of height to shadow length should be equal for the building and the pole. So, \(\frac{h}{20}=\frac{4}{4}\) (Wait, there's a typo? Wait, the pole is 4 - meter tall and shadow 4 meters? Wait, no, maybe a typo. Wait, maybe the pole is 4 - meter tall and shadow 5 meters? Wait, no, the original problem: "a 4 - meter pole casts a shadow 4 meters long"? Wait, no, maybe it's a 5 - meter pole? Wait, no, the options are 20, 15, 16, 10. Wait, maybe the pole is 5 meters? Wait, no, let's re - check. Wait, maybe the pole is 5 meters? Wait, no, the problem says "a 4 - meter pole casts a shadow 4 meters long"? No, that would make the ratio 1. Then the building's height would be 20. But that seems odd. Wait, maybe it's a 5 - meter pole? Wait, no, the problem as given: "A building casts a shadow 20 meters long, and a 4 - meter pole casts a shadow 4 meters long. What is the height of the building?" Wait, if the pole is 4m tall and shadow 4m, then the ratio of height to shadow is \( \frac{4}{4}=1 \). So the building's height \( h \) has \( \frac{h}{20}=1 \), so \( h = 20 \). But let's check the options. Option a is 20 meters.

Step2: Solve the proportion

From \(\frac{h}{20}=\frac{4}{4}\), since \(\frac{4}{4}=1\), then \( h=20\times1 = 20 \).

Step1: Recall similarity ratio

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{AB}{DE}=\frac{AC}{DF}\).

Step2: Substitute values

We know \( AB = 10 \), \( AC = 15 \), \( DE = 5 \). Substitute into the proportion: \(\frac{10}{5}=\frac{15}{DF}\).

Step3: Solve for \( DF \)

Simplify \(\frac{10}{5}=2\), so \( 2=\frac{15}{DF} \). Cross - multiply: \( 2\times DF=15 \), then \( DF=\frac{15}{2}=7.5 \).

Step1: Analyze the triangles

The triangles formed by the flagpole (height \( h_1 \), shadow \( s_1 = 24 \)) and the pole (height \( h_2 = 3 \), shadow \( s_2 = 4 \)) are right triangles (since the objects are vertical and shadows are horizontal, so the angle between the object and the ground is \( 90^{\circ} \)). Also, the angle of the sun is the same for both, so the non - right angles are equal (corresponding angles).

Step2: Determine the criterion

In two right triangles, if two angles are equal (the right angle and the angle from the sun), by the AA (Angle - Angle) similarity criterion, the triangles are similar. The AA criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. Here, both triangles are right - angled (one angle equal) and they share the same angle of elevation from the sun (second angle equal). So the AA Criterion confirms the similarity.

Answer:

a. 20 meters

Second Question (Similar Triangles \( \triangle ABC\sim\triangle DEF \))