Constructing Inscribed and Circumscribed Circles - Geometry
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If $AC$ is a diameter and $B$ is on the circle, what is $m\angle ABC$?
If $AC$ is a diameter and $B$ is on the circle, what is $m\angle ABC$?
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$90^\circ$. Thales' theorem: angles in semicircle are $90^\circ$.
$90^\circ$. Thales' theorem: angles in semicircle are $90^\circ$.
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What is a cyclic quadrilateral?
What is a cyclic quadrilateral?
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A quadrilateral with all four vertices on one circle. Also called an inscribed quadrilateral.
A quadrilateral with all four vertices on one circle. Also called an inscribed quadrilateral.
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Which construction lines locate a triangle's incenter?
Which construction lines locate a triangle's incenter?
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Angle bisectors of at least two angles. Two lines are sufficient since they intersect at one point.
Angle bisectors of at least two angles. Two lines are sufficient since they intersect at one point.
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What is the circumcircle of a triangle?
What is the circumcircle of a triangle?
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The circle through all three vertices of the triangle. The circumcenter is the center with circumradius as radius.
The circle through all three vertices of the triangle. The circumcenter is the center with circumradius as radius.
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What is the incircle of a triangle?
What is the incircle of a triangle?
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The circle tangent to all three sides of the triangle. The incenter is the center with inradius as radius.
The circle tangent to all three sides of the triangle. The incenter is the center with inradius as radius.
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What is a perpendicular bisector of a segment?
What is a perpendicular bisector of a segment?
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A line perpendicular to the segment at its midpoint. This creates two congruent right triangles.
A line perpendicular to the segment at its midpoint. This creates two congruent right triangles.
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What is an angle bisector?
What is an angle bisector?
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A ray that divides an angle into two congruent angles. This creates two angles of equal measure.
A ray that divides an angle into two congruent angles. This creates two angles of equal measure.
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What point is equidistant from all three vertices of a triangle?
What point is equidistant from all three vertices of a triangle?
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The circumcenter. The circumcenter is where perpendicular bisectors meet.
The circumcenter. The circumcenter is where perpendicular bisectors meet.
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What point is equidistant from all three sides of a triangle?
What point is equidistant from all three sides of a triangle?
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The incenter. The incenter is where angle bisectors meet.
The incenter. The incenter is where angle bisectors meet.
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Where is the circumcenter located in an acute triangle?
Where is the circumcenter located in an acute triangle?
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Inside the triangle. All angles are less than $90^\circ$ in acute triangles.
Inside the triangle. All angles are less than $90^\circ$ in acute triangles.
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Where is the circumcenter located in a right triangle?
Where is the circumcenter located in a right triangle?
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At the midpoint of the hypotenuse. The hypotenuse is a diameter of the circumcircle.
At the midpoint of the hypotenuse. The hypotenuse is a diameter of the circumcircle.
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Where is the circumcenter located in an obtuse triangle?
Where is the circumcenter located in an obtuse triangle?
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Outside the triangle. One angle exceeds $90^\circ$ in obtuse triangles.
Outside the triangle. One angle exceeds $90^\circ$ in obtuse triangles.
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Where is the incenter located in any nondegenerate triangle?
Where is the incenter located in any nondegenerate triangle?
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Inside the triangle. The incenter always lies within the triangle's interior.
Inside the triangle. The incenter always lies within the triangle's interior.
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What is the defining property of points on a perpendicular bisector of $\overline{AB}$?
What is the defining property of points on a perpendicular bisector of $\overline{AB}$?
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They are equidistant from $A$ and $B$. This is the perpendicular bisector theorem.
They are equidistant from $A$ and $B$. This is the perpendicular bisector theorem.
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What is the defining property of points on the angle bisector of $\angle A$?
What is the defining property of points on the angle bisector of $\angle A$?
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They are equidistant from the sides of $\angle A$. This is the angle bisector theorem property.
They are equidistant from the sides of $\angle A$. This is the angle bisector theorem property.
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Identify the circle used to circumscribe $\triangle ABC$ once the circumcenter $O$ is found.
Identify the circle used to circumscribe $\triangle ABC$ once the circumcenter $O$ is found.
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Circle centered at $O$ with radius $OA$. All vertices must lie on this circle.
Circle centered at $O$ with radius $OA$. All vertices must lie on this circle.
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Identify the circle used to inscribe in $\triangle ABC$ once the incenter $I$ is found.
Identify the circle used to inscribe in $\triangle ABC$ once the incenter $I$ is found.
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Circle centered at $I$ with radius equal to the distance to a side. The circle must be tangent to all three sides.
Circle centered at $I$ with radius equal to the distance to a side. The circle must be tangent to all three sides.
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What is the relationship between an inscribed angle and its intercepted arc?
What is the relationship between an inscribed angle and its intercepted arc?
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Inscribed angle measure is half the intercepted arc measure. This is the inscribed angle theorem.
Inscribed angle measure is half the intercepted arc measure. This is the inscribed angle theorem.
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What is the measure of an inscribed angle intercepting a $100^\circ$ arc?
What is the measure of an inscribed angle intercepting a $100^\circ$ arc?
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$50^\circ$. Apply the inscribed angle theorem: $\frac{100^\circ}{2} = 50^\circ$.
$50^\circ$. Apply the inscribed angle theorem: $\frac{100^\circ}{2} = 50^\circ$.
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What is the arc measure intercepted by an inscribed angle of $35^\circ$?
What is the arc measure intercepted by an inscribed angle of $35^\circ$?
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$70^\circ$. Apply the inscribed angle theorem: $35^\circ \times 2 = 70^\circ$.
$70^\circ$. Apply the inscribed angle theorem: $35^\circ \times 2 = 70^\circ$.
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What is the relationship between a central angle and its intercepted arc?
What is the relationship between a central angle and its intercepted arc?
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Central angle measure equals intercepted arc measure. Central angles directly measure their intercepted arcs.
Central angle measure equals intercepted arc measure. Central angles directly measure their intercepted arcs.
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What is the measure of a central angle intercepting a $120^\circ$ arc?
What is the measure of a central angle intercepting a $120^\circ$ arc?
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$120^\circ$. Central angle equals its intercepted arc measure.
$120^\circ$. Central angle equals its intercepted arc measure.
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What is a cyclic quadrilateral?
What is a cyclic quadrilateral?
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A quadrilateral with all four vertices on one circle. Also called an inscribed quadrilateral.
A quadrilateral with all four vertices on one circle. Also called an inscribed quadrilateral.
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State the opposite-angle property of a cyclic quadrilateral $ABCD$.
State the opposite-angle property of a cyclic quadrilateral $ABCD$.
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$m\angle A + m\angle C = 180^\circ$ and $m\angle B + m\angle D = 180^\circ$. Opposite angles are supplementary in cyclic quadrilaterals.
$m\angle A + m\angle C = 180^\circ$ and $m\angle B + m\angle D = 180^\circ$. Opposite angles are supplementary in cyclic quadrilaterals.
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If $ABCD$ is cyclic and $m\angle A = 68^\circ$, what is $m\angle C$?
If $ABCD$ is cyclic and $m\angle A = 68^\circ$, what is $m\angle C$?
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$112^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 68^\circ = 112^\circ$.
$112^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 68^\circ = 112^\circ$.
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If $ABCD$ is cyclic and $m\angle D = 103^\circ$, what is $m\angle B$?
If $ABCD$ is cyclic and $m\angle D = 103^\circ$, what is $m\angle B$?
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$77^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 103^\circ = 77^\circ$.
$77^\circ$. Opposite angles sum to $180^\circ$: $180^\circ - 103^\circ = 77^\circ$.
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What condition on angles proves a quadrilateral is cyclic?
What condition on angles proves a quadrilateral is cyclic?
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A pair of opposite angles is supplementary. This is the converse of the cyclic quadrilateral theorem.
A pair of opposite angles is supplementary. This is the converse of the cyclic quadrilateral theorem.
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Identify whether $ABCD$ is cyclic if $m\angle A = 92^\circ$ and $m\angle C = 88^\circ$.
Identify whether $ABCD$ is cyclic if $m\angle A = 92^\circ$ and $m\angle C = 88^\circ$.
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Yes, because $92^\circ + 88^\circ = 180^\circ$. Opposite angles are supplementary, confirming it's cyclic.
Yes, because $92^\circ + 88^\circ = 180^\circ$. Opposite angles are supplementary, confirming it's cyclic.
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Identify whether $ABCD$ is cyclic if $m\angle B = 95^\circ$ and $m\angle D = 70^\circ$.
Identify whether $ABCD$ is cyclic if $m\angle B = 95^\circ$ and $m\angle D = 70^\circ$.
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No, because $95^\circ + 70^\circ \ne 180^\circ$. The sum is $165^\circ$, not $180^\circ$, so not cyclic.
No, because $95^\circ + 70^\circ \ne 180^\circ$. The sum is $165^\circ$, not $180^\circ$, so not cyclic.
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What is the tangent-chord angle theorem (angle formed by tangent and chord)?
What is the tangent-chord angle theorem (angle formed by tangent and chord)?
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The angle equals half the measure of its intercepted arc. This applies to angles formed by tangent and chord.
The angle equals half the measure of its intercepted arc. This applies to angles formed by tangent and chord.
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