Question

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

Answer 1

Step-by-step 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 \\ √(11)=CP \end{gathered}`

The value of CP is:

`\boxed{CP=√(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=(5)/(6)`

Solving for C:

`C=sin^(-1)((5)/(6))`

Solving the operations:

Answer: