To find the length of segment \( HE \), we can use the intersecting chords theorem, which states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.
In this problem, we have the following information:
- Chord \( FG \) with segments \( FE = 12 \) and \( EG = 7 \)
- Chord \( HK \) with segments \( HE = x \) (unknown) and \( EK = 21 \)
According to the intersecting chords theorem:
\[ FE \cdot EG = HE \cdot EK \]
Substituting the known values into the equation gives:
\[ 12 \cdot 7 = x \cdot 21 \]
Calculating the left side:
\[ 84 = x \cdot 21 \]
To find \( x \), we can divide both sides by 21:
\[ x = \frac{84}{21} = 4 \]
Thus, the length of segment \( HE \) is 4 units.
The correct response is:
Segment HE is 4 units long.
1 answer