To find the possible base and height of a second triangle that has the same area as the first triangle, we can start by calculating the area of the first triangle.
The area \(A\) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the first triangle, with a base of \(12\) and height of \(8\):
\[ A = \frac{1}{2} \times 12 \times 8 = \frac{1}{2} \times 96 = 48 \]
Now, we know that the second triangle must also have an area of \(48\).
Since the base varies inversely with the height, we can express this relationship mathematically. Let \(b\) be the base and \(h\) be the height of the second triangle. The inverse relationship can be represented as:
\[ b = \frac{k}{h} \]
for some constant \(k\). Since the area of the second triangle is also 48, we have:
\[ \frac{1}{2} \times b \times h = 48 \]
Multiplying both sides by \(2\) gives:
\[ b \times h = 96 \]
Now we can substitute \(b = \frac{k}{h}\) into the area equation:
\[ \frac{k}{h} \times h = 96 \]
This simplifies to:
\[ k = 96 \]
Now, we can express the base \(b\) in terms of height \(h\):
\[ b = \frac{96}{h} \]
Thus, for any height \(h\), the base \(b\) is given by this equation. The possible pairs \((b, h)\) of base and height that maintain the area of \(48\) are:
- If \(h = 1\), then \(b = 96\).
- If \(h = 2\), then \(b = 48\).
- If \(h = 3\), then \(b = 32\).
- If \(h = 4\), then \(b = 24\).
- If \(h = 6\), then \(b = 16\).
- If \(h = 8\), then \(b = 12\).
- If \(h = 12\), then \(b = 8\).
- If \(h = 16\), then \(b = 6\).
- If \(h = 24\), then \(b = 4\).
- If \(h = 32\), then \(b = 3\).
- If \(h = 48\), then \(b = 2\).
- If \(h = 96\), then \(b = 1\).
These pairs show how the base must decrease as the height increases while maintaining the same area of \(48\).
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