Question

PQ is tangent to •C at P. If PQ = 5 and CQ = 6, find CP and m

Answer 1

Answer:

$\begin{gathered} CP=\sqrt{11} \\ m\operatorname{\angle}C=56.44 \end{gathered}$

Explanation:

Step 1. The information that we have is that

• PQ=5

,

• CQ=6,

and that PQ is tangent to circle C.

Since PQ is a tangent line, it forms a 90° angle with the circumference, and the triangle is a right triangle.

We need to find CP and the measure of angle C (m

Step 2. To find CP we use the Pythagorean theorem:

In this case:

$(CQ)^2=(CP)^2+(PQ)^2$

Substituting the known values:

$6^2=(CP)^2+5^2$

Solving for CP:

$\begin{gathered} 6^2-5^2=(CP)^2 \\ 36-25=(CP)^2 \\ 11=(CP)^2 \\ \sqrt{11}=CP \end{gathered}$

The value of CP is:

$\boxed{CP=\sqrt{11}}$

Step 3. To find the measure of angle C, we use the trigonometric function sine:

$sinC=\frac{opposite\text{ side}}{hypotenuse}$

The opposite side to angle C is 5 and the hypotenuse is 6:

$sinC=\frac{5}{6}$

Solving for C:

$C=sin^{-1}(\frac{5}{6})$

Solving the operations:

$\begin{gathered} C=s\imaginaryI n^{-1}(0.83333) \\ C=56.44 \\ \downarrow \\ \boxed{m\operatorname{\angle}C=56.44} \end{gathered}$

Answer:

$\begin{gathered} CP=\sqrt{11} \\ m\operatorname{\angle}C=56.44 \end{gathered}$