Question 1210480: Find the ratio of the area of the red region to the area of the yellow region. Enter your answer as a fraction.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2103)   (Show Source):
You can put this solution on YOUR website! This is a trick question! The description is inconsistent.
* If the **red region is a triangle** and the **yellow region is the same triangle**, their areas must be **equal**.
* If the **yellow region is the same triangle, half as big**, the yellow region cannot be the *same* triangle. It must be a **similar triangle** whose dimensions are scaled down.
Let's proceed with the most logical interpretation for a geometry/ratio problem: the yellow triangle is **similar** to the red triangle, with its linear dimensions scaled by a factor of $1/2$.
## 📐 Ratio of Areas for Similar Figures
When two geometric figures are **similar** (the same shape but different sizes), the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions (e.g., sides, heights).
Let:
* $A_R$ be the area of the **red triangle**.
* $A_Y$ be the area of the **yellow triangle**.
* $s_R$ be a linear dimension (side length) of the red triangle.
* $s_Y$ be the corresponding linear dimension of the yellow triangle.
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### Step 1: Determine the Ratio of Linear Dimensions
The problem states the yellow triangle is "half as big." In geometry, "half as big" usually means the linear dimensions are halved:
$$\frac{s_Y}{s_R} = \frac{1}{2}$$
### Step 2: Determine the Ratio of Areas
The ratio of the areas of two similar figures is the square of the ratio of their corresponding linear dimensions:
$$\frac{A_Y}{A_R} = \left(\frac{s_Y}{s_R}\right)^2$$
$$\frac{A_Y}{A_R} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$
### Step 3: Find the Required Ratio
The question asks for the ratio of the area of the **red region** to the area of the **yellow region** ($\frac{A_R}{A_Y}$):
$$\frac{A_R}{A_Y} = \frac{1}{\frac{A_Y}{A_R}} = \frac{1}{\frac{1}{4}} = \mathbf{4}$$
The ratio of the area of the red region to the area of the yellow region is **$\frac{4}{1}$**.
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If you strictly follow the first part of the statement ("The red region is a triangle, and the yellow region is the **same triangle**"):
* $A_R = A_Y$
* Ratio $\frac{A_R}{A_Y} = \frac{1}{1}$
However, this makes the second part ("half as big") irrelevant or contradictory. The answer $\frac{4}{1}$ is the correct solution based on the intended mathematical meaning of a scaled figure.
Do you have another geometry problem you'd like to solve?
Answer by ikleyn(53250)  (Show Source):
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