Question

QS
is tangent to ⨀P. What is PS?
P
Q
S
4 yd
3 yd
PS=
yd

Answer 1

To find the length of \( PS \) when \( QS \) is tangent to the circle (denoted by \( \odot P \)) at point \( S \), we can use the property of tangents.

The length of a tangent from a point outside the circle to the point of tangency is equal to the square root of the distance from the center of the circle to the point outside the circle minus the radius of the circle.

Given:
- \( PQ = 4 \) yards (the distance from point \( P \) to point \( Q \))
- \( PS = 3 \) yards (the radius of the circle at point \( S \))

Using the properties of a right triangle formed by the radius \( PS \), the tangent \( QS \), and the line \( PQ \):
1. We know that \( PQ^2 = PS^2 + QS^2 \) (from the Pythagorean theorem).

Now substituting the values we have:
- \( PQ = 4 \) yards
- \( PS = 3 \) yards

Let’s find \( QS \):
\[
PQ^2 = PS^2 + QS^2
\]
\[
4^2 = 3^2 + QS^2
\]
\[
16 = 9 + QS^2
\]
\[
QS^2 = 16 - 9
\]
\[
QS^2 = 7
\]
\[
QS = \sqrt{7} \text{ yards}
\]

To find \( PS \) which was already given as \( 3 \) yards, the question might be ambiguous, as we already know \( PS \).

Thus, in conclusion:
\[
PS = 3 \text{ yards}
\]