The question asks how many times greater the area of Triangle B is compared to Triangle A, given that the base and height of Triangle A are half the base and height of Triangle B.

The area of a triangle is calculated using the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Let’s define:

• Base of Triangle A = $\frac{1}{2} \times$ Base of Triangle B
• Height of Triangle A = $\frac{1}{2} \times$ Height of Triangle B

Calculating the area ratio:

1. The area of Triangle A is: $\text{Area of A} = \frac{1}{2} \times \left(\frac{\text{Base of B}}{2}\right) \times \left(\frac{\text{Height of B}}{2}\right) = \frac{1}{2} \times \frac{1}{4} \times \text{Base of B} \times \text{Height of B} = \frac{1}{8} \times \text{Base of B} \times \text{Height of B}$

2. The area of Triangle B is: $\text{Area of B} = \frac{1}{2} \times \text{Base of B} \times \text{Height of B}$

3. The ratio of the areas is: $\text{Area of B} : \text{Area of A} = \left(\frac{1}{2} \times \text{Base of B} \times \text{Height of B}\right) \div \left(\frac{1}{8} \times \text{Base of B} \times \text{Height of B}\right) = 8$

Thus, the area of Triangle B is 8 times greater than the area of Triangle A.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What happens to the area if only the base of Triangle A is halved, but the height remains the same?
2. How does the area of a triangle change when both dimensions are scaled by a factor of 3?
3. Can you determine the ratio of areas for triangles with other scale factors between their dimensions?
4. How would the result differ if the shapes were not triangles but rectangles?
5. What other geometric properties are affected by scaling the dimensions of a shape?

Tip: When both the base and height of a shape are scaled by a factor of $n$, the area is scaled by $n^2$.