Answer:
The length of the arc = [tex]114.66\,cm[/tex]
And the length of the full circle: [tex]306.3\, cm[/tex]
Step-by-step explanation:
To solve this problem we will be analyzing aright angle triangle that is formed when drawing the appropriate radius associated with the arc and the chord of a segment of circumference as depicted by the triangle shaded in yellow in the attached image.
Notice that the information given in this problem can be represented by:
1) the 90 cm width of the arc with the value of 2x (drawn in orange) in the drawing
2) the 30 cm of the arc's height with the value of y (drawn in green) in the drawing
In order to use the Pythagorean theorem in the little yellow triangle, notice that the hypotenuse is given by the radius R of the circle, and the two legs of the right angle triangle are given by "x" and "R-y"
So the Pythagoras theorem becomes:
[tex]R^2=x^2+(R-y)^2\\R^2=x^2+R^2-2Ry+y^2\\0=x^2-2Ry+y^2\\2Ry=x^2+y^2\\R=\frac{x^2+y^2}{2y}[/tex]
Therefore, in our case, where x=45 cm, and y=30 cm, we have:
[tex]R=\frac{x^2+y^2}{2y} \\R=\frac{45^2+30^2}{60}\,cm \\R=48.75\, cm[/tex]
Now, with the radius, we can calculate the length of the arc of circle if we know the subtended angle in radians:
length of arc = [tex]R \,\theta[/tex]
The angle [tex]\theta[/tex] subtended by the arc can be obtained by doubling the angle with vertex at the circle's center in our yellow right angle triangle. This is done via the sine function:
[tex]\theta=2\,arcsin(x/R)\\\theta=2\,arcsin(45/48.75)\\\theta=2.352\,radians[/tex]
Therefore, length of the arc = [tex]R \,\theta=48.75\,cm\,*\,2.352=114.66\,cm[/tex]
And the length of the full circle: [tex]2\,\pi \, R =2\,\pi\,48.75\,cm = 306.3\, cm[/tex]
