Question 3

Question

Fill in the blank question.
△RVT
is similar to △SUT
. What is the measure of side UT
? Enter the answer as a decimal.

Triangle SUT inside similar triangle RVT; RV is 10, VT is 32, SU is 6, UT is unknown

UT=

Answer 1

To find the length of side UT in triangle SUT, given that triangles RVT and SUT are similar, we can set up a proportion based on the corresponding sides of the similar triangles.

The sides of triangle RVT are:

• RV = 10
• VT = 32

The sides of triangle SUT are:

• SU = 6
• UT = unknown (let's call it x)

Since the triangles are similar, the ratio of the sides must be the same. We can set up the proportion as follows:

\[ \frac{SU}{RV} = \frac{UT}{VT} \]

Substituting the known values into the proportion gives us:

\[ \frac{6}{10} = \frac{x}{32} \]

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

\[ 6 \cdot 32 = 10 \cdot x \]

Calculating \(6 \cdot 32\):

\[ 192 = 10x \]

Now, divide both sides by 10 to solve for \(x\):

\[ x = \frac{192}{10} = 19.2 \]

So, the measure of side \(UT\) is:

\[ UT = 19.2 \]

Therefore, the answer is:

UT = 19.2

Answer 2

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? Enter the answer

Answer 3

To find the height of the child using the concept of similar triangles, we can set up a proportion based on the flagpole and the child's shadow.

Let's denote the height of the child as \( h \).

The given information is:

• 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 formed by the flagpole and its shadow and the child and their shadow are similar, we can set up the following proportion:

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

Substituting in the values we have:

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

Now, cross-multiplying gives:

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

Calculating the left side:

\[ 60 = 20h \]

Now, divide both sides by 20 to solve for \( h \):

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

Thus, the height of the child is:

3 feet.