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We are given a banner composed of two congruent triangles and a rectangle. The dimensions provided are:

- Rectangle's width: 17 cm
- Rectangle's height: 10 cm
- The base of the triangle: 12 cm (corresponds to the triangle’s height since it's right-angled)

### Step 1: Calculate the area of the rectangle.
The area of a rectangle is given by the formula:

\[
A_{\text{rectangle}} = \text{width} \times \text{height}
\]

For the rectangle:
\[
A_{\text{rectangle}} = 17 \times 10 = 170 \, \text{cm}^2
\]

### Step 2: Calculate the area of one triangle.
The area of a triangle is given by:

\[
A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height}
\]

For one triangle:
\[
A_{\text{triangle}} = \frac{1}{2} \times 12 \times 10 = 60 \, \text{cm}^2
\]

Since there are two congruent triangles:
\[
A_{\text{triangles}} = 2 \times 60 = 120 \, \text{cm}^2
\]

### Step 3: Total area of the original banner.
The total area of the banner (before dilation) is the sum of the areas of the rectangle and the two triangles:

\[
A_{\text{total original}} = 170 + 120 = 290 \, \text{cm}^2
\]

### Step 4: Effect of dilation by \( \frac{1}{2} \).
When an object is dilated by a factor of \( \frac{1}{2} \), its area scales by the square of the dilation factor:

\[
\text{New area} = (\text{dilation factor})^2 \times \text{original area}
\]

For a dilation factor of \( \frac{1}{2} \):
\[
\text{New area} = \left(\frac{1}{2}\right)^2 \times 290 = \frac{1}{4} \times 290 = 72.5 \, \text{cm}^2
\]

### Final Answer:
The total area of the banner after dilation is **72.5 cm²**.

Do you want more details or have any questions? Here are some related questions for further understanding:

1. What would the total area be if the dilation factor was \( \frac{1}{3} \)?
2. How does dilation affect the perimeter of the banner?
3. If the banner was dilated by a factor of 2, what would the new area be?
4. Can you calculate the area of just one triangle after the dilation?
5. How would you calculate the volume if this shape were extended into a three-dimensional prism?

**Tip:** Dilation affects both area and perimeter, but in different ways. Area scales by the square of the dilation factor, while the perimeter scales linearly by the factor itself.

If the banner is dilated by 1/2, what is the total area?

Calculate the total area of a banner composed of two congruent triangles and a rectangle, using area formulas and the effect of dilation by 1/2. Learn how the dilation factor affects the area and review key geometry concepts. Suitable for Grades 8-10.

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