Similar Shapes: Find x cm Height for 25 cm² Area

Answer

The value of $x$ is 7.5 cm. By using the relationship between the areas of similar shapes, we determined the scale factor for the lengths and applied it to the known height.

Explanation

The image shows two heart-shaped objects that are mathematically similar, meaning they have the same shape but different sizes. When shapes are similar, their areas scale by the square of their length ratio.

1. Identify the known quantities

   • Area of larger shape ($A_{large}$) = $36\text{ cm}^2$
   • Area of smaller shape ($A_{small}$) = $25\text{ cm}^2$
   • Height of larger shape ($H_{large}$) = $9\text{ cm}$
   • Height of smaller shape ($H_{small}$) = $x\text{ cm}$

2. Establish the Ratio of Areas The ratio of the areas of two similar shapes is equal to the square of the ratio of their corresponding lengths (like height). Let $k$ be the scale factor for lengths: $\frac{A_{small}}{A_{large}} = k^2 = \frac{25}{36}$ This formula shows that the area ratio is the square of the ratio of the two heights.

3. Find the Length Scale Factor ($k$) To find $k$, we take the square root of the area ratio: $k = \sqrt{\frac{25}{36}} = \frac{5}{6}$ This tells us that every length on the small heart is $\frac{5}{6}$ the size of the corresponding length on the large heart.

4. Solve for $x$ Now we apply the scale factor $k$ to the height of the larger shape: $x = H_{large} \times k = 9 \times \frac{5}{6}$ We multiply the large height by the length ratio to find the smaller unknown height. $x = \frac{45}{6} = 7.5$ This is the final calculation step to find the value of the unknown height.

5. Unit Check We started with cm for height and cm$^2$ for area. After calculating the square root of the area ratio, we are left with a unitless ratio, which we applied to the cm height measurement, resulting in an answer in cm.

Final Answer

$\boxed{7.5}$

Common Mistakes

• Mixing up the ratio: Students sometimes calculate $\frac{36}{25}$ instead of $\frac{25}{36}$. Always check that if the target shape is smaller, your scale factor is less than 1.
• Forgetting the square root: A very common error is to assume the ratio of heights is the same as the ratio of areas ($x/9 = 25/36$). Remember: lengths scale linearly, but areas scale by the square of the factor.