Question

Find the values of x in the figurePQ is tangent to .o at q-30-60-40-50

Answer 1

Since the line segment PQ is tantgent to the circle, then the angle mTherefore we know that the angle OQR has to be 30°. This is shown in the figure:

Now, we know that the sum of all the internal angles of a triangle must be 180°, then for the triangle RQP we have:

`30+\beta+x=180`

For the triangle OQR we have

`\alpha+\gamma+60=180`

and for the triangle OQP we have

`x+\alpha+90=180`

Summing up we have the system

`\begin{gathered} x+\beta+30=180 \\ \alpha+\gamma+60=180 \\ x+\alpha+90=180 \end{gathered}`

At first glance we see that this system hava more unknown variables than equations, but we have to notice that the angles alpha and gamma are supplementary, then