Question
UV
is tangent to ⨀T. What is TV?
T
U
V
4 m
3 m
TV=
m
is tangent to ⨀T. What is TV?
T
U
V
4 m
3 m
TV=
m
Answers
GPT-4o mini
To find the length of \( TV \), we can use the fact that a tangent line to a circle forms a right triangle with the radius drawn to the point of tangency.
In this scenario, we have:
- \( TU \) as the radius of the circle, which is \( 4 \, m \).
- \( UV \) as the length of the tangent that is \( 3 \, m \).
- \( TV \) is the length we want to find.
In triangle \( TUV \), we apply the Pythagorean theorem:
\[
TV^2 + UV^2 = TU^2
\]
Substituting the known values:
\[
TV^2 + 3^2 = 4^2
\]
This simplifies to:
\[
TV^2 + 9 = 16
\]
Subtract \( 9 \) from both sides:
\[
TV^2 = 16 - 9
\]
\[
TV^2 = 7
\]
Taking the square root of both sides gives:
\[
TV = \sqrt{7} \approx 2.65 \, m
\]
Therefore, the length of \( TV \) is
\[
TV \approx 2.65 \, m \quad (\text{exactly } \sqrt{7} \, m).
\]
In this scenario, we have:
- \( TU \) as the radius of the circle, which is \( 4 \, m \).
- \( UV \) as the length of the tangent that is \( 3 \, m \).
- \( TV \) is the length we want to find.
In triangle \( TUV \), we apply the Pythagorean theorem:
\[
TV^2 + UV^2 = TU^2
\]
Substituting the known values:
\[
TV^2 + 3^2 = 4^2
\]
This simplifies to:
\[
TV^2 + 9 = 16
\]
Subtract \( 9 \) from both sides:
\[
TV^2 = 16 - 9
\]
\[
TV^2 = 7
\]
Taking the square root of both sides gives:
\[
TV = \sqrt{7} \approx 2.65 \, m
\]
Therefore, the length of \( TV \) is
\[
TV \approx 2.65 \, m \quad (\text{exactly } \sqrt{7} \, m).
\]