To find the value of \( x \), we can compare the areas of the two triangles, since they both share the concept of base and height.
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the first triangle with a height of 12 inches and a base of \( x - 7 \):
\[ A_1 = \frac{1}{2} \times (x - 7) \times 12 \]
For the second triangle with a height of 8 inches and a base of 14 inches:
\[ A_2 = \frac{1}{2} \times 14 \times 8 \]
Now, let's calculate \( A_2 \):
\[ A_2 = \frac{1}{2} \times 14 \times 8 = 7 \times 8 = 56 , \text{square inches} \]
Now, set the areas equal to each other since we want to find \( x \):
\[ \frac{1}{2} \times (x - 7) \times 12 = 56 \]
Multiply both sides by 2 to eliminate the fraction:
\[ (x - 7) \times 12 = 112 \]
Now divide both sides by 12:
\[ x - 7 = \frac{112}{12} \]
Simplifying the right side:
\[ x - 7 = \frac{28}{3} \]
Now add 7 to both sides:
\[ x = 7 + \frac{28}{3} = \frac{21}{3} + \frac{28}{3} = \frac{49}{3} \]
To find the numerical value, convert \( \frac{49}{3} \) to a decimal:
\[ x \approx 16.33 \]
Since none of the given options matches \( \frac{49}{3} \), it seems that we need to reconsider our approach or clarify the equality condition.
To check if \( x \) can be one of the given options, let's substitute each option back into the area equation and see which, if any, results in equal areas:
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If \( x = 24 \):
- Base: \( 24 - 7 = 17 \)
- Area: \( A_1 = \frac{1}{2} \times 17 \times 12 = 102 \) (not equal to 56)
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If \( x = 28 \):
- Base: \( 28 - 7 = 21 \)
- Area: \( A_1 = \frac{1}{2} \times 21 \times 12 = 126 \) (not equal to 56)
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If \( x = 20 \):
- Base: \( 20 - 7 = 13 \)
- Area: \( A_1 = \frac{1}{2} \times 13 \times 12 = 78 \) (not equal to 56)
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If \( x = 32 \):
- Base: \( 32 - 7 = 25 \)
- Area: \( A_1 = \frac{1}{2} \times 25 \times 12 = 150 \) (not equal to 56)
None of these options yield equal areas when plugged into the first triangle's area formula.
Under the initial interpretations and verifications of the equations based on equal area principles, we confirm that \( x = \frac{49}{3} \) or approximately \( 16.33 \) stands correct, but the answer does not align with the listed \( x \) values provided.
Therefore, it seems there may be a misunderstanding or an error with the triangle conditions or given options. If the question has specific constraints or if we're overlooking specific equality conditions in the two triangles, reevaluating those might be necessary. Otherwise, based on area calculations, the value derived doesn't align with listed options.
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