Pentagon FGHJK is similar to pentagon MPQST

Question

Pentagon FGHJK is 8 cm in height the width of the pentagon is 12 cm with a letter on every point of the pentagon
Pentagon MPQST is 12 cm with x cm width
What is the value of x?
Set up a proportion to solve for x. Show how you solved the proportion by using cross products, setting up an equation, and isolating the variable.

Answer 1

To find the width \( x \) of pentagon MPQST, we can use the property of similar figures, which states that the ratios of corresponding linear dimensions in similar figures are equal.

In this case, we know:

• The height of pentagon FGHJK = 8 cm
• The width of pentagon FGHJK = 12 cm
• The height of pentagon MPQST = 12 cm
• The width of pentagon MPQST = \( x \) cm

Since the pentagons are similar, we can write the proportion of their heights and their widths as follows:

\[ \frac{\text{Height of FGHJK}}{\text{Height of MPQST}} = \frac{\text{Width of FGHJK}}{\text{Width of MPQST}} \]

Substituting the known values into the proportion:

\[ \frac{8}{12} = \frac{12}{x} \]

Next, we can use cross products to solve for \( x \):

\[ 8 \cdot x = 12 \cdot 12 \]

Calculating the right side:

\[ 8x = 144 \]

Now, isolate \( x \) by dividing both sides by 8:

\[ x = \frac{144}{8} \]

Calculating the value of \( x \):

\[ x = 18 \]

Thus, the width \( x \) of pentagon MPQST is \( \boxed{18} \) cm.