If the height of a triangle is halved and the area of a triangle is doubled, then

Select one:

a.

the base is halved.

b.

the base is one fourth of the original.

c.

the base is doubled.

d.

the base is four times of the original.

To solve this problem, let's analyze the relationships between the area, base, and height of the triangle.

### Initial Area of the Triangle:
The area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Let's denote the original base as \( b \) and the original height as \( h \). Therefore, the original area \( A_1 \) is:
\[
A_1 = \frac{1}{2} \times b \times h
\]

### New Area of the Triangle:
According to the problem, the height is halved and the area is doubled.

- New height = \( \frac{h}{2} \)
- New area = \( 2A_1 \)

Let the new base be \( b' \). The new area \( A_2 \) can be expressed as:
\[
A_2 = \frac{1}{2} \times b' \times \frac{h}{2}
\]
But we know \( A_2 = 2A_1 \), so:
\[
\frac{1}{2} \times b' \times \frac{h}{2} = 2 \times \frac{1}{2} \times b \times h
\]

Simplifying this equation:
\[
\frac{1}{4} \times b' \times h = b \times h
\]

Divide both sides by \( h \):
\[
\frac{b'}{4} = b
\]

Finally, multiply both sides by 4:
\[
b' = 4b
\]

### Conclusion:
The new base \( b' \) is four times the original base \( b \). Therefore, the correct answer is:

**d. the base is four times of the original.**

Would you like further details or have any other questions? Here are some related questions to consider:

1. What happens to the area of the triangle if both the base and height are doubled?
2. How would the area change if only the base of a triangle is tripled?
3. If the area of a triangle remains constant but the base is doubled, what happens to the height?
4. How does the area change if the height is increased by 50%?
5. Can you derive the formula for the area of a triangle using its three sides?

**Tip:** Always keep in mind how changes to base or height directly affect the area of a triangle, given their proportional relationship.

Explore how halving the height and doubling the area of a triangle affects its base. Detailed solution provided with formulas and step-by-step analysis. Suitable for Grades 9-12.

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