Pentagon fghjk is similar to pentagon mpqst . 8 cm in between k and f and 12 cm in between j and h and 12 in between t and m and x cm in s and q on the second pentagon. What is the value of x? Set up a proportion to solve for x. Show how you solved the proportion by using cross products, selling up an equation, and isolating the variable.

Question

GPT-4o mini · Answer

To find the value of $ x $ in the similar pentagons $ fghjk $ and $ mpqst $, we can use the property of similar polygons, which states that the ratios of corresponding sides are equal. Given: - The length between $ k $ and $ f $ is $ 8 $ cm. - The length between $ j $ and $ h $ is $ 12 $ cm. - The length between $ t $ and $ m $ is $ 12 $ cm. - The length between $ s $ and $ q $ is $ x $ cm. We can set up the proportion based on the corresponding sides of the similar pentagons: $$ \frac{kf}{tm} = \frac{jh}{sq} $$ Substituting the known values into the proportion: $$ \frac{8}{12} = \frac{12}{x} $$ Now, we can cross-multiply to solve for $ x $: $$ 8x = 12 \cdot 12 $$ Calculate $ 12 \cdot 12 $: $$ 8x = 144 $$ Next, isolate $ x $ by dividing both sides by $ 8 $: $$ x = \frac{144}{8} $$ Perform the division: $$ x = 18 $$ Thus, the length $ x $ between $ s $ and $ q $ in pentagon $ mpqst $ is $ 18 $ cm. In summary, we set up the proportion: 1. $ \frac{8}{12} = \frac{12}{x} $ 2. Cross multiplied to get $ 8x = 144 $ 3. Divided by $ 8 $ to solve for $ x $: $ x = 18 $ cm.