Question

Use the diagram to the right to find x.

a.
470
b. 29°
C.) 181°

Answer 1

`\large\boxed{\tt x = 47^(\circ).}`

Explanation:

`\textsf{We are asked to find the value of x.}`

`\textsf{We are given an angle with an arc measurement in between 2 \underline{chords}.}`

`\large\underline{\textsf{What is a Chord?}}`

`\textsf{A Chord is a line segment whose endpoints are \underline{on} the circle.}`

`\textsf{For our problem, we have 2 Chords that \underline{Intersect}.}`

`\textsf{Because the Intersection Point (Vertex) is inside the circle, we can identify x}`

`\textsf{using what we are given.}`

`\textsf{Intersecting Chords create an angle that is equal to half the sum of the 2 arcs.}`

`\underline{\textsf{Formatted into an equation;}}`

`\tt Angle = (1)/(2) (Arc \ 1 + Arc \ 2)`

`\textsf{We are given 2 out of 3 of these values, let's substitute them inside the equation.}`

`\tt 76^(\circ) = (1)/(2) (105^(\circ) + x \ (Arc \ 2))`

`\large\underline{\textsf{Solving;}}`

`\textsf{We should solve for x by simplifying the equation. Let's attempt to remove} \ \tt (1)/(2)`

`\textsf{from the equation by multiplying both sides of the equation by the reciprocal of}`

`\tt (1)/(2). \ \textsf{Lastly, simplify then divide both sides of the equation by 2.}`

`\large\underline{\textsf{What is a Reciprocal?}}`

`\textsf{A Reciprocal is a fraction where the Numerator and Denominator are switched.}`

`\underline{\textsf{Multiply both sides of the equation by 2;}}`

`\textsf{2 is the reciprocal of} \ \tt (1)/(2) .`

`\tt 2 * 76^(\circ) = 2 * (1)/(2) (105^(\circ) + x)`

`\tt 152^(\circ) = 105^(\circ) + x`

`\underline{\textsf{Subtract 105 from both sides of the equation;}}`

`\tt 152^(\circ) - 105^(\circ)= 105^(\circ) - 105^(\circ) + x`

`\large\underline{\textsf{Hence;}}`

`\large\boxed{\tt x = 47^(\circ).}`