Question

are the expressions for the side lengths possible for this equiangular triangle?

yes
no

Explanation:

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<reasoning>

Equate side lengths

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &14x - 4 = 4x + 1\\ &10x = 5\\ &x = 0.5 \end{aligned}$$

\]

Verify consistency of the third side

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &\text{Side 1: } 14(0.5) - 4 = 7 - 4 = 3\\ &\text{Side 2: } 4(0.5) + 1 = 2 + 1 = 3\\ &\text{Side 3: } 12(0.5) - 3 = 6 - 3 = 3 \end{aligned}$$

\]

Determine possibility

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &\text{All three sides are equal to } 3.\\ &\text{Since } 3 > 0\text{, these side lengths are geometrically possible.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>Yes</mcq-correct>
<mcq-option>No</mcq-option>
</answer>

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Answer:

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<reasoning>

Equate side lengths

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &14x - 4 = 4x + 1\\ &10x = 5\\ &x = 0.5 \end{aligned}$$

\]

Verify consistency of the third side

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &\text{Side 1: } 14(0.5) - 4 = 7 - 4 = 3\\ &\text{Side 2: } 4(0.5) + 1 = 2 + 1 = 3\\ &\text{Side 3: } 12(0.5) - 3 = 6 - 3 = 3 \end{aligned}$$

\]

Determine possibility

Using the Equilateral Triangle Properties knowledge point
\[

$$\begin{aligned} &\text{All three sides are equal to } 3.\\ &\text{Since } 3 > 0\text{, these side lengths are geometrically possible.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>Yes</mcq-correct>
<mcq-option>No</mcq-option>
</answer>

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