Question

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i is the midpoint of \\(\overline{hj}\\) and g is the midpoint of \\(\overline{hk}\\). if \\(jk = s\\) and \\(gi = s - 27\\), what is the value of \\(s\\)?

Explanation:

Step1: Apply Midline Theorem

In triangle \( \triangle HJK \), \( G \) is the midpoint of \( HK \) and \( I \) is the midpoint of \( HJ \). By the Midline Theorem (also known as the Midsegment Theorem), the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long. So, \( GI \parallel JK \) and \( GI=\frac{1}{2}JK \).

Step2: Substitute Given Values

We know \( JK = s \) and \( GI = s - 27 \). From the Midline Theorem, we have the equation:

$$ s - 27=\frac{1}{2}s $$

Step3: Solve for \( s \)

Subtract \( \frac{1}{2}s \) from both sides:

$$ s-\frac{1}{2}s - 27=0 $$

$$ \frac{1}{2}s - 27=0 $$

Add 27 to both sides:

$$ \frac{1}{2}s=27 $$

Multiply both sides by 2:

$$ s = 54 $$

Answer:

\( 54 \)