Question

In the diagram below of circle C, mQT = 140, and m

Answer 1

Given:

Angle P is formed by two secants intersecting outside of circle C.

The intercepted arcs are major arc QT and minor arc RS.

$\hat{mQT}=140\degree\text{ and }m\angle P=40\degree$

Required:

$\text{ We need to find }m\mathring{RS}.$

Explanation:

Recall that

$\text{ Angle Formed by Two Secants =}\frac{1}{2}(\text{ \lparen Difference of Intercepted Arcs\rparen}$$m\angle P=\frac{1}{2}(m\hat{QT}-m\hat{RS})$$Substitute\text{ }\hat{mQT}=140\degree\text{ and }m\angle P=40\degree\text{ in the equation.}$$40\degree=\frac{1}{2}(140\degree-m\hat{RS})$

Multiply both sides by 2.

$2\times40\degree=2\times\frac{1}{2}(140\degree-m\hat{RS})$$80\degree=140\degree-m\hat{RS}$$Add\text{ }m\hat{RS}-80\degree\text{ on both sides of the equation.}$$80\degree+m\hat{RS}-80\degree=140\degree-m\hat{RS}+m\hat{RS}-80\degree$$m\hat{RS}=140\degree-80\degree$$m\hat{RS}=60\degree$

Final answer:

$m\hat{RS}=60\degree$