Constructing Tangents to Circles - Geometry
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What must be true about a line $\ell$ through $T$ to be tangent to circle with center $O$?
What must be true about a line $\ell$ through $T$ to be tangent to circle with center $O$?
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$\ell \perp OT$. Perpendicularity to the radius defines tangency.
$\ell \perp OT$. Perpendicularity to the radius defines tangency.
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Identify the points that determine the tangent lines in the standard construction from external point $P$.
Identify the points that determine the tangent lines in the standard construction from external point $P$.
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The tangency points $T_1$ and $T_2$ where circles intersect. These points lie on both circles and create tangent lines to $P$.
The tangency points $T_1$ and $T_2$ where circles intersect. These points lie on both circles and create tangent lines to $P$.
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What is the geometric reason the constructed lines $PT_1$ and $PT_2$ touch the circle only once each?
What is the geometric reason the constructed lines $PT_1$ and $PT_2$ touch the circle only once each?
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Each is perpendicular to the corresponding radius at $T_1$ or $T_2$. The perpendicular relationship ensures single-point contact.
Each is perpendicular to the corresponding radius at $T_1$ or $T_2$. The perpendicular relationship ensures single-point contact.
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What is the value of $PT$ if $OP=25$ and the circle radius is $r=7$?
What is the value of $PT$ if $OP=25$ and the circle radius is $r=7$?
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$24$. Using $PT = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$.
$24$. Using $PT = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$.
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What tool-based action constructs the midpoint $M$ of $OP$ in a compass-straightedge construction?
What tool-based action constructs the midpoint $M$ of $OP$ in a compass-straightedge construction?
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Construct the perpendicular bisector of $OP$ to locate $M$. Use compass arcs to find the midpoint geometrically.
Construct the perpendicular bisector of $OP$ to locate $M$. Use compass arcs to find the midpoint geometrically.
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What theorem explains why the midpoint-circle construction produces tangency points?
What theorem explains why the midpoint-circle construction produces tangency points?
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An angle inscribed in a semicircle is a right angle. Also known as Thales' theorem, ensuring $\angle OTP = 90^\circ$.
An angle inscribed in a semicircle is a right angle. Also known as Thales' theorem, ensuring $\angle OTP = 90^\circ$.
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Identify whether two tangents exist if $OP=11$ and $r=11$ for circle centered at $O$.
Identify whether two tangents exist if $OP=11$ and $r=11$ for circle centered at $O$.
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One tangent. When $OP = r$, point $P$ lies on the circle.
One tangent. When $OP = r$, point $P$ lies on the circle.
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In the construction, what is the measure of $\angle OTP$ when $T$ is a tangency point?
In the construction, what is the measure of $\angle OTP$ when $T$ is a tangency point?
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$90^\circ$. The tangent creates a right angle with the radius.
$90^\circ$. The tangent creates a right angle with the radius.
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What segment becomes a diameter of the auxiliary circle used in the tangent construction?
What segment becomes a diameter of the auxiliary circle used in the tangent construction?
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$OP$ is a diameter of circle centered at midpoint $M$. Using $OP$ as diameter ensures right angles at intersection points.
$OP$ is a diameter of circle centered at midpoint $M$. Using $OP$ as diameter ensures right angles at intersection points.
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What is the value of $PT$ if $OP=10$ and the circle radius is $r=6$?
What is the value of $PT$ if $OP=10$ and the circle radius is $r=6$?
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$8$. Using $PT = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$.
$8$. Using $PT = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$.
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What is the key right triangle formed when constructing tangents from $P$ to a circle centered at $O$?
What is the key right triangle formed when constructing tangents from $P$ to a circle centered at $O$?
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Right triangle $\triangle OTP$ with right angle at $T$. The tangent perpendicularity creates this right triangle.
Right triangle $\triangle OTP$ with right angle at $T$. The tangent perpendicularity creates this right triangle.
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What equality holds for the two tangent segments from the same external point $P$ to a circle?
What equality holds for the two tangent segments from the same external point $P$ to a circle?
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$PT_1=PT_2$. Both tangent segments from the same external point are equal.
$PT_1=PT_2$. Both tangent segments from the same external point are equal.
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What is the name of the theorem stating $PT_1=PT_2$ for tangents from external point $P$?
What is the name of the theorem stating $PT_1=PT_2$ for tangents from external point $P$?
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Tangent segments from a common external point are congruent. This theorem follows from the symmetry of the construction.
Tangent segments from a common external point are congruent. This theorem follows from the symmetry of the construction.
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Which circle do you draw to use Thales' theorem when constructing tangents from $P$ to circle $(O,r)$?
Which circle do you draw to use Thales' theorem when constructing tangents from $P$ to circle $(O,r)$?
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The circle with diameter $OP$. This circle has $OP$ as its diameter.
The circle with diameter $OP$. This circle has $OP$ as its diameter.
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What is the required input information to construct a tangent from an external point to a circle?
What is the required input information to construct a tangent from an external point to a circle?
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A circle (center and radius) and an external point $P$. You need the circle specification and external point location.
A circle (center and radius) and an external point $P$. You need the circle specification and external point location.
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What is the standard auxiliary circle center when constructing tangents from $P$ to circle centered at $O$?
What is the standard auxiliary circle center when constructing tangents from $P$ to circle centered at $O$?
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The midpoint $M$ of $OP$. Using the midpoint creates the proper auxiliary circle.
The midpoint $M$ of $OP$. Using the midpoint creates the proper auxiliary circle.
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What is the standard auxiliary circle radius when constructing tangents from $P$ to circle centered at $O$?
What is the standard auxiliary circle radius when constructing tangents from $P$ to circle centered at $O$?
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$MO$ (which equals $MP$). Both distances equal half of $OP$.
$MO$ (which equals $MP$). Both distances equal half of $OP$.
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What is the length relationship between $MO$ and $MP$ when $M$ is the midpoint of $OP$?
What is the length relationship between $MO$ and $MP$ when $M$ is the midpoint of $OP$?
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$MO=MP$. Since $M$ is the midpoint, both distances are equal.
$MO=MP$. Since $M$ is the midpoint, both distances are equal.
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What must be true about $OP$ to ensure two distinct tangents exist from $P$ to circle $(O,r)$?
What must be true about $OP$ to ensure two distinct tangents exist from $P$ to circle $(O,r)$?
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$OP>r$. $P$ must be outside the circle for two tangents to exist.
$OP>r$. $P$ must be outside the circle for two tangents to exist.
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What is the result if $OP=r$ when attempting to construct tangents from $P$ to circle $(O,r)$?
What is the result if $OP=r$ when attempting to construct tangents from $P$ to circle $(O,r)$?
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There is exactly one tangent at $P$. Point $P$ lies on the circle, creating one tangent.
There is exactly one tangent at $P$. Point $P$ lies on the circle, creating one tangent.
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What is the result if $OP<r$ when attempting to construct tangents from $P$ to circle $(O,r)$?
What is the result if $OP<r$ when attempting to construct tangents from $P$ to circle $(O,r)$?
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No tangent from $P$ exists. Point $P$ is inside the circle, preventing tangent construction.
No tangent from $P$ exists. Point $P$ is inside the circle, preventing tangent construction.
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What is the measure of the angle between a tangent and the radius drawn to the tangency point?
What is the measure of the angle between a tangent and the radius drawn to the tangency point?
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$90^\circ$. Tangent lines are always perpendicular to radii at tangency points.
$90^\circ$. Tangent lines are always perpendicular to radii at tangency points.
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Identify the correct construction step: after drawing $OP$, what do you construct next to find tangency points?
Identify the correct construction step: after drawing $OP$, what do you construct next to find tangency points?
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Construct midpoint $M$ of $OP$. The midpoint enables the auxiliary circle construction.
Construct midpoint $M$ of $OP$. The midpoint enables the auxiliary circle construction.
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Identify the correct construction step: after finding midpoint $M$ of $OP$, what do you do next?
Identify the correct construction step: after finding midpoint $M$ of $OP$, what do you do next?
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Draw the circle centered at $M$ passing through $O$. This circle passes through both $O$ and $P$.
Draw the circle centered at $M$ passing through $O$. This circle passes through both $O$ and $P$.
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Identify the correct construction step: after drawing the circle with diameter $OP$, what do you mark?
Identify the correct construction step: after drawing the circle with diameter $OP$, what do you mark?
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Its intersections $T_1,T_2$ with the given circle. These are the tangency points for the construction.
Its intersections $T_1,T_2$ with the given circle. These are the tangency points for the construction.
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Identify the correct construction step: after marking $T_1$ and $T_2$, what do you draw?
Identify the correct construction step: after marking $T_1$ and $T_2$, what do you draw?
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Lines $PT_1$ and $PT_2$. These are the required tangent lines from $P$.
Lines $PT_1$ and $PT_2$. These are the required tangent lines from $P$.
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What is the tangent length formula from external point $P$ to tangency point $T$ in terms of $OP$ and $r$?
What is the tangent length formula from external point $P$ to tangency point $T$ in terms of $OP$ and $r$?
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$PT=\sqrt{OP^2-r^2}$. Derived from the Pythagorean theorem in right triangle $\triangle OTP$.
$PT=\sqrt{OP^2-r^2}$. Derived from the Pythagorean theorem in right triangle $\triangle OTP$.
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What equation expresses the right triangle relationship in $\triangle OTP$ for a tangent from $P$?
What equation expresses the right triangle relationship in $\triangle OTP$ for a tangent from $P$?
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$OP^2=OT^2+PT^2$. This is the Pythagorean theorem applied to the tangent triangle.
$OP^2=OT^2+PT^2$. This is the Pythagorean theorem applied to the tangent triangle.
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What is $OT$ equal to if $T$ is a tangency point on a circle with radius $r$ and center $O$?
What is $OT$ equal to if $T$ is a tangency point on a circle with radius $r$ and center $O$?
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$OT=r$. The radius equals the distance from center to any point on the circle.
$OT=r$. The radius equals the distance from center to any point on the circle.
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What is the value of $PT$ if $OP=13$ and the circle radius is $r=5$?
What is the value of $PT$ if $OP=13$ and the circle radius is $r=5$?
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$12$. Using $PT = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
$12$. Using $PT = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$.
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