Question
QS
is tangent to ⨀P. What is PS?
P
Q
S
15 km
8 km
PS=
km
is tangent to ⨀P. What is PS?
P
Q
S
15 km
8 km
PS=
km
Answers
GPT-4o mini
To find the length of PS when \( QS \) is tangent to the circle (denoted as \( \odot P \)), we can use the property that a tangent to a circle is perpendicular to the radius at the point of tangency.
Given the information:
- \( PQ = 15 \) km (the radius of the circle)
- \( QS = 8 \) km (length of the tangent)
We can apply the Pythagorean theorem in the triangle \( PQS \):
\[
PQ^2 + QS^2 = PS^2
\]
Substituting the known values:
\[
15^2 + 8^2 = PS^2
\]
Calculating \( 15^2 \) and \( 8^2 \):
\[
225 + 64 = PS^2
\]
Adding these together:
\[
289 = PS^2
\]
Now, take the square root of both sides to find \( PS \):
\[
PS = \sqrt{289} = 17 \text{ km}
\]
Therefore, \( PS = 17 \) km.
Given the information:
- \( PQ = 15 \) km (the radius of the circle)
- \( QS = 8 \) km (length of the tangent)
We can apply the Pythagorean theorem in the triangle \( PQS \):
\[
PQ^2 + QS^2 = PS^2
\]
Substituting the known values:
\[
15^2 + 8^2 = PS^2
\]
Calculating \( 15^2 \) and \( 8^2 \):
\[
225 + 64 = PS^2
\]
Adding these together:
\[
289 = PS^2
\]
Now, take the square root of both sides to find \( PS \):
\[
PS = \sqrt{289} = 17 \text{ km}
\]
Therefore, \( PS = 17 \) km.