randRange( 0, 360 )
randRange( 10, 80 ) * 2
( START + 180 ) % 360
4
CENTRAL
$._("blue")
$._("orange")
$._("green angle")
$._("pink angle")
$._("orange angle")
If the GIVEN_LABEL angle measures GIVEN degrees, what does the ASKED_LABEL angle measure?
This is a special case where the blue and orange angles' sides share the same line. The blue angle is called a central angle, and the orange angle is called an inscribed 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
The green and blue angles are supplementary. Because the blue angle is CENTRAL degrees, the green angle must be 180 - CENTRAL degrees.
style({ stroke: GREEN }, function() {
graph.circle.drawCentralArc( START + CENTRAL, SUBTENDED_POINT );
})
We know that the angles in a triangle sum to 180 degrees.
style({ stroke: PINK }, function() {
graph.circle.drawInscribedArc( START + CENTRAL, SUBTENDED_POINT, START + CENTRAL + 180 );
});
\color{GREEN}{\text{GREEN_ANGLE}} + \color{PINK}{\text{PINK_ANGLE}} + \color{ORANGE}{\text{ORANGE_ANGLE}} = 180^{\circ}
The pink sides of the triangle are radii, so they must be equal.
style({ stroke: PINK }, function() {
graph.circle.drawRadius( START + CENTRAL );
graph.circle.drawRadius( SUBTENDED_POINT );
});
This means that the triangle is isosceles and that the base angles, or the pink and orange angles, are equal.
\color{GREEN}{\text{GREEN_ANGLE}} + 2 \cdot \color{ORANGE}{\text{ORANGE_ANGLE}} = 180^{\circ}
2 \cdot \color{ORANGE}{\text{ORANGE_ANGLE}} = 180^{\circ} - \color{GREEN}{180 - CENTRAL^{\circ}}
2 \cdot \color{ORANGE}{\text{ORANGE_ANGLE}} = \color{BLUE}{CENTRAL^{\circ}}
\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")
$._("green angle")
$._("pink angle")
$._("orange angle")
$._("blue angle")
CENTRAL degrees
The pink sides of the triangle are radii, so they must be equal. This means the triangle is isosceles and that the base angles, or the pink and orange angles, are equal.
style({ stroke: PINK }, function() {
graph.circle.drawRadius( START + CENTRAL );
graph.circle.drawRadius( SUBTENDED_POINT );
});
style({ stroke: PINK }, function() {
graph.circle.drawInscribedArc( START + CENTRAL, SUBTENDED_POINT, START + CENTRAL + 180 );
});
We know that the angles in a triangle sum to 180 degrees.
style({ stroke: GREEN }, function() {
graph.circle.drawCentralArc( START + CENTRAL, SUBTENDED_POINT );
})
\color{GREEN}{\text{GREEN_ANGLE}} + \color{PINK}{\text{PINK_ANGLE}} + \color{ORANGE}{\text{ORANGE_ANGLE}} = 180^{\circ}
The green and blue angles are supplementary.
\color{GREEN}{\text{GREEN_ANGLE}} + \color{BLUE}{\text{BLUE_ANGLE}} = 180^{\circ}
\color{BLUE}{\text{BLUE_ANGLE}} = 180^{\circ} - \color{GREEN}{\text{GREEN_ANGLE}}
\color{BLUE}{\text{BLUE_ANGLE}} = 180^{\circ} - \color{GREEN}{\text{180 - CENTRAL}^{\circ}}
\color{BLUE}{\text{BLUE_ANGLE}} = \color{BLUE}{CENTRAL^{\circ}}
randRange( 0, 360 )
randRange( 10, 80 ) * 2
randRange( START + 180 + 5, START + CENTRAL + 180 - 5 ) % 360
4
CENTRAL
$._("blue")
$._("orange")
$._("orange angle")
$._("blue angle")
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 inside 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.central = graph.circle.drawCentralAngle( START, START + CENTRAL );
style({ stroke: ORANGE, fill: ORANGE });
graph.inscribed = 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 sub-angles formed by the dashed diameter shown above?
style({stroke: BLUE, "stroke-dasharray": "-"}, function() {
graph.circle.drawChord( SUBTENDED_POINT, SUBTENDED_POINT + 180 );
});
If we only look at the sub-angles drawn now, we see that this is the special case from the previous inscribed angles exercise!
We know that the orange sub-angle is one half the measure of the blue sub-angle.
graph.inscribed.arc.animate({opacity: 0.4});
graph.central.arc.animate({opacity: 0.4});
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( SUBTENDED_POINT + 180, START + CENTRAL, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( SUBTENDED_POINT, SUBTENDED_POINT + 180, START + CENTRAL, 0.7 );
Likewise, the other orange sub-angle is one half the measure of the other blue sub-angle, as shown.
graph.centralSub.remove();
graph.inscribedSub.remove();
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( START, SUBTENDED_POINT + 180, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( SUBTENDED_POINT, START, SUBTENDED_POINT + 180, 0.7 );
If both orange sub-angles are one half both blue sub-angles, then we know that the original orange angle is one half the original blue angle.
graph.centralSub.remove();
graph.inscribedSub.remove();
graph.inscribed.arc.animate({opacity: 1.0});
graph.central.arc.animate({opacity: 1.0});
\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")
$._("blue angle")
$._("orange angle")
CENTRAL degrees
What do we know about the sub-angles formed by the dashed diameter shown above?
style({stroke: BLUE, "stroke-dasharray": "-"}, function() {
graph.circle.drawChord( SUBTENDED_POINT, SUBTENDED_POINT + 180 );
});
If we only look at the sub-angles drawn now, we see that this is the special case from the previous inscribed angles exercise!
We know that the orange sub-angle is one half the measure of the blue sub-angle.
graph.inscribed.arc.animate({opacity: 0.4});
graph.central.arc.animate({opacity: 0.4});
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( SUBTENDED_POINT + 180, START + CENTRAL, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( SUBTENDED_POINT, SUBTENDED_POINT + 180, START + CENTRAL, 0.7 );
Likewise, the other orange sub-angle is one half the measure of the other blue sub-angle, as shown.
graph.centralSub.remove();
graph.inscribedSub.remove();
style({stroke: BLUE});
graph.centralSub = graph.circle.drawCentralArc( START, SUBTENDED_POINT + 180, 0.7 );
style({stroke: ORANGE});
graph.inscribedSub = graph.circle.drawInscribedArc( SUBTENDED_POINT, START, SUBTENDED_POINT + 180, 0.7 );
If both blue sub-angles are twice both blue sub-angles, then we know that the blue is twice the orange angle.
graph.centralSub.remove();
graph.inscribedSub.remove();
graph.inscribed.arc.animate({opacity: 1.0});
graph.central.arc.animate({opacity: 1.0});
\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}}
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")
$._("green angle")
$._("blue angle")
$._("pink angle")
$._("small green angle")
$._("orange angle")
$._("big green angle")
$._("small pink angle")
$._("big pink angle")
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}}