Question

Naveed was given a circle with the chords FG
and HK
that intersect at point E
inside the circle. If the length of FE=12
, EG=7
, and EK=21
, investigate this circle and find the length of segment HE
.(1 point)
Responses

Answer 1

To find the length of segment \( HE \), we can use the property of intersecting chords in a circle. According to this property, when two chords intersect, the products of the lengths of the segments of each chord are equal.

In this case, we can denote the segments as follows:

- For chord \( FG \):
- \( FE = 12 \)
- \( EG = 7 \)

Thus, the product of the segments of chord \( FG \) is:
\[
FE \times EG = 12 \times 7 = 84.
\]

- For chord \( HK \):
- \( EK = 21 \)
- \( HE = x \) (we are trying to find \( HE \))

Thus, the product of the segments of chord \( HK \) is:
\[
EK \times HE = 21 \times x.
\]

According to the intersecting chords theorem:
\[
FE \times EG = EK \times HE,
\]
which gives us the equation:
\[
84 = 21 \times x.
\]

Now, we can solve for \( x \):
\[
x = \frac{84}{21} = 4.
\]

Thus, the length of segment \( HE \) is \( \boxed{4} \).