A flagpole casts a 20-foot shadow at the same time a child casts a 5-foot shadow. If the flagpole is 12 feet tall, how tall is the child?

Answer 1

To determine the height of the child using the information given, we can apply the principle of similar triangles. The flagpole and its shadow form one triangle, and the child and his shadow form another triangle.

Let \( h \) be the height of the child. We have the following information:

• Height of the flagpole: \( 12 \) feet
• Length of the flagpole's shadow: \( 20 \) feet
• Length of the child's shadow: \( 5 \) feet

Since the triangles are similar, we can set up a proportion based on the heights and the lengths of the shadows:

\[ \frac{\text{Height of Flagpole}}{\text{Length of Flagpole's Shadow}} = \frac{\text{Height of Child}}{\text{Length of Child's Shadow}} \]

Substituting the known values into the proportion gives us:

\[ \frac{12}{20} = \frac{h}{5} \]

Now, we can cross-multiply to solve for \( h \):

\[ 12 \cdot 5 = 20 \cdot h \]

This simplifies to:

\[ 60 = 20h \]

Next, we can divide both sides by \( 20 \):

\[ h = \frac{60}{20} = 3 \]

Therefore, the height of the child is \( \boxed{3} \) feet.

Answer 2

Two students use different proportions to find the unknown length of side d
of the similar triangles.

Similar right triangles that share a vertex; smaller triangle has height 2 and base 6; larger triangle has height 5 and base d

Student A sets up the proportion 26=5d
, while student B sets up the proportion 25=6d
. Which student’s proportion is correct? What is the measure of the unknown side length? Choose the correct answers.

1 of 2.
Select Choice
is/are correct.

d=
2 of 2.
Select Choice

Answer 3

To determine which student's proportion is correct, we need to analyze their setups with respect to the similar triangles.

Given:

• Smaller Triangle:

  • Height = 2
  • Base = 6

• Larger Triangle:

  • Height = 5
  • Base = \( d \)

Since the triangles are similar, the ratios of corresponding sides are equal. This means we can set up the proportions correctly as follows:

1. Correct Proportion Based on Heights and Bases: \[ \frac{\text{Height of Smaller Triangle}}{\text{Height of Larger Triangle}} = \frac{\text{Base of Smaller Triangle}}{\text{Base of Larger Triangle}} \]

This gives us: \[ \frac{2}{5} = \frac{6}{d} \]

Solving for \( d \):

Cross-multiply to solve for \( d \): \[ 2d = 5 \times 6 \]

Calculating: \[ 2d = 30 \] \[ d = \frac{30}{2} = 15 \]

Thus, the correct value for \( d \) is \( 15 \).

Examining Students’ Proportions:

• Student A: Sets up the proportion \( 26 = 5d \). This does not make sense because \( 26 \) is not a correct ratio based on the triangle dimensions given.

• Student B: Sets up the proportion \( 25 = 6d \), which is also incorrect based on the triangle dimensions.

Thus, neither student has the correct proportion set up.

Answers:

1. The correct proportions are not given by either student.
2. The measure of the unknown side length \( d \) is \( 15 \).

Final Answers:

1 of 2. None of the students are correct. 2 of 2. d = 15.