Question
Segments Inside a Circle Quick Check
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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
Segment HE
is 25 units long.
Segment upper H upper E is 25 units long.
Segment HE
is 4 units long.
Segment upper H upper E is 4 units long.
Segment HE
is 19 units long.
Segment upper H upper E is 19 units long.
Segment HE
is 84 units long.
Segment upper H upper E is 84 units long.
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Answers
GPT-4o mini
To find the length of segment HE, we can utilize the intersecting chords theorem, which states that if two chords intersect each other inside a circle, then the products of their segments are equal.
Given:
- \( FE = 12 \)
- \( EG = 7 \)
- \( EK = 21 \)
Let \( HE = x \).
According to the theorem, we have:
\[
FE \cdot EG = HE \cdot EK
\]
Substituting the known values:
\[
12 \cdot 7 = x \cdot 21
\]
Calculating the left side:
\[
84 = x \cdot 21
\]
Now, solve for \( x \):
\[
x = \frac{84}{21} = 4
\]
So, the length of segment HE is \( 4 \) units.
The correct response is:
**Segment HE is 4 units long. Segment HE is 4 units long.**
Given:
- \( FE = 12 \)
- \( EG = 7 \)
- \( EK = 21 \)
Let \( HE = x \).
According to the theorem, we have:
\[
FE \cdot EG = HE \cdot EK
\]
Substituting the known values:
\[
12 \cdot 7 = x \cdot 21
\]
Calculating the left side:
\[
84 = x \cdot 21
\]
Now, solve for \( x \):
\[
x = \frac{84}{21} = 4
\]
So, the length of segment HE is \( 4 \) units.
The correct response is:
**Segment HE is 4 units long. Segment HE is 4 units long.**