randRange( 0, 360 ) randRange( 10, 80 ) * 2 randFromArray([ randRange( START + CENTRAL + 5, START + 180 - 5 ), randRange( START + CENTRAL + 180 + 5, START + 360 - 5 ) ]) % 360 4 CENTRAL "blue" "orange"

If the GIVEN_LABEL angle measures GIVEN degrees, what does the ASKED_LABEL angle measure?

This is a special case where the center of the circle is outside the inscribed orange angle. The blue angle is called a central angle.

init({ range: [ [ -RADIUS - 1, RADIUS + 1 ], [ -RADIUS - 1, RADIUS + 1 ] ] }); addMouseLayer(); graph.circle = new Circle( RADIUS ); style({ stroke: BLUE, fill: BLUE }); graph.circle.drawCenter(); graph.circle.drawPoint( START ); graph.circle.drawPoint( START + CENTRAL ); graph.circle.drawCentralAngle( START, START + CENTRAL ); style({ stroke: ORANGE, fill: ORANGE }); graph.circle.drawInscribedAngle( SUBTENDED_POINT, START, START + CENTRAL ); graph.circle.drawMovablePoint( SUBTENDED_POINT, START + CENTRAL, START );
CENTRAL / 2 degrees

What do we know about the angles formed by the dashed diameter shown above?

style({stroke: BLUE, "stroke-dasharray": "-"}, function() { graph.circle.drawChord( SUBTENDED_POINT, SUBTENDED_POINT + 180 ); });

From the previous inscribed angles exercises, we know the following about the green and pink angles.

\color{GREEN}{\text{green angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{pink angle}}

style({stroke: BLUE, fill: BLUE}, function() { graph.circle.drawPoint( SUBTENDED_POINT + 180 ); }); style({stroke: PINK}); var arc = innerArc( START, ( SUBTENDED_POINT + 180 ) % 360 ); graph.central = graph.circle.drawCentralAngle( arc.start, arc.end, 0.7 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( SUBTENDED_POINT, arc.start, arc.end, 0.7 );

We can see another pair of these special case inscribed and central angles, with the same relationship between green and pink angles.

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove(); var arc = innerArc( START + CENTRAL, ( SUBTENDED_POINT + 180 ) % 360 ); style({stroke: PINK}); graph.central = graph.circle.drawCentralAngle( arc.start, arc.end, 0.9 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( SUBTENDED_POINT, arc.start, arc.end, 0.9 );

Looking at the picture, we can see the following is true:

\color{GREEN}{\text{small green angle}} + \color{ORANGE}{\text{orange angle}} = \color{GREEN}{\text{big green angle}}

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove();

Substituting what we know about green and pink angles, we get the following:

\dfrac{1}{2} \cdot \color{PINK}{\text{small pink angle}} + \color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{PINK}{\text{big pink angle}}

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2}( \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}})

We can see from the picture that the following is also true:

\color{PINK}{\text{small pink angle}} + \color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}}

\color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}} - \color{PINK}{\text{small pink angle}}

Combining what we know about blue and orange angles:

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{\text{blue angle}}

\color{ORANGE}{\text{orange angle}} = \dfrac{1}{2} \cdot \color{BLUE}{CENTRAL^{\circ}}

\color{ORANGE}{\text{orange angle}} = \color{ORANGE}{CENTRAL / 2^{\circ}}

CENTRAL / 2 "orange" "blue"
CENTRAL degrees

What do we know about the angles formed by the dashed diameter shown above?

style({stroke: BLUE, "stroke-dasharray": "-"}, function() { graph.circle.drawChord( SUBTENDED_POINT, SUBTENDED_POINT + 180 ); });

From the previous inscribed angles exercises, we know the following about the green and pink angles.

\color{PINK}{\text{pink angle}} = 2 \cdot \color{GREEN}{\text{green angle}}

style({stroke: BLUE, fill: BLUE}, function() { graph.circle.drawPoint( SUBTENDED_POINT + 180 ); }); style({stroke: PINK}); var arc = innerArc( START, ( SUBTENDED_POINT + 180 ) % 360 ); graph.central = graph.circle.drawCentralAngle( arc.start, arc.end, 0.7 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( SUBTENDED_POINT, arc.start, arc.end, 0.7 );

We can see another pair of these special case inscribed and central angles, with the same relationship between green and pink angles.

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove(); var arc = innerArc( START + CENTRAL, ( SUBTENDED_POINT + 180 ) % 360 ); style({stroke: PINK}); graph.central = graph.circle.drawCentralAngle( arc.start, arc.end, 0.9 ); style({stroke: GREEN}); graph.inscribed = graph.circle.drawInscribedAngle( SUBTENDED_POINT, arc.start, arc.end, 0.9 );

Looking at the picture, we can see the following is true:

\color{PINK}{\text{small pink angle}} + \color{BLUE}{\text{blue angle}} = \color{PINK}{\text{big pink angle}}

graph.central.arc.animate({opacity: 0.4}); graph.central.radii[0].remove(); graph.central.radii[1].remove(); graph.inscribed.arc.animate({opacity: 0.4}); graph.inscribed.chords[0].remove(); graph.inscribed.chords[1].remove();

Substituting what we know about green and pink angles, we get the following:

2 \cdot \color{GREEN}{\text{small green angle}} + \color{BLUE}{\text{blue angle}} = 2 \cdot \color{GREEN}{\text{big green angle}}

\color{BLUE}{\text{blue angle}} = 2( \color{GREEN}{\text{big green angle}} - \color{GREEN}{\text{small green angle}})

We can see from the picture that the following is also true:

\color{GREEN}{\text{small green angle}} + \color{ORANGE}{\text{orange angle}} = \color{GREEN}{\text{big green angle}}

\color{ORANGE}{\text{orange angle}} = \color{GREEN}{\text{big green angle}} - \color{GREEN}{\text{small green angle}}

Combining what we know about blue and orange angles:

\color{BLUE}{\text{blue angle}} = 2 \cdot \color{ORANGE}{\text{orange angle}}

\color{BLUE}{\text{blue angle}} = 2 \cdot \color{ORANGE}{CENTRAL / 2^{\circ}}

\color{BLUE}{\text{blue angle}} = \color{BLUE}{CENTRAL^{\circ}}