Constructing Tangents to Circles - Pre-Calculus
Card 1 of 30
Find the tangent length: a circle has $r=5$ and an external point has $OP=13$; what is $PT$?
Find the tangent length: a circle has $r=5$ and an external point has $OP=13$; what is $PT$?
Tap to reveal answer
$12$. Use $PT=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}$.
$12$. Use $PT=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}$.
← Didn't Know|Knew It →
Find the tangent length: $r=6$ and $OP=10$; what is $PT$?
Find the tangent length: $r=6$ and $OP=10$; what is $PT$?
Tap to reveal answer
$8$. Use $PT=\sqrt{10^2-6^2}=\sqrt{100-36}=\sqrt{64}$.
$8$. Use $PT=\sqrt{10^2-6^2}=\sqrt{100-36}=\sqrt{64}$.
← Didn't Know|Knew It →
Identify the number of tangents: if $OP=9$ and $r=4$, how many tangents from $P$ exist?
Identify the number of tangents: if $OP=9$ and $r=4$, how many tangents from $P$ exist?
Tap to reveal answer
$2$. Since $OP>r$, point $P$ is external, allowing two tangents.
$2$. Since $OP>r$, point $P$ is external, allowing two tangents.
← Didn't Know|Knew It →
Identify the number of tangents: if $OP=7$ and $r=7$, how many tangents from $P$ exist?
Identify the number of tangents: if $OP=7$ and $r=7$, how many tangents from $P$ exist?
Tap to reveal answer
$1$. When $OP=r$, point $P$ is on the circle with one tangent.
$1$. When $OP=r$, point $P$ is on the circle with one tangent.
← Didn't Know|Knew It →
Identify the number of tangents: if $OP=3$ and $r=5$, how many real tangents from $P$ exist?
Identify the number of tangents: if $OP=3$ and $r=5$, how many real tangents from $P$ exist?
Tap to reveal answer
$0$. Since $OP<r$, point $P$ is inside the circle with no tangents.
$0$. Since $OP<r$, point $P$ is inside the circle with no tangents.
← Didn't Know|Knew It →
What is the tangent line to $x^2 + y^2 = 16$ at $(0,4)$?
What is the tangent line to $x^2 + y^2 = 16$ at $(0,4)$?
Tap to reveal answer
$y = 4$. Horizontal tangent at top of circle (vertical radius).
$y = 4$. Horizontal tangent at top of circle (vertical radius).
← Didn't Know|Knew It →
What theorem relates a tangent line to the radius at the point of tangency?
What theorem relates a tangent line to the radius at the point of tangency?
Tap to reveal answer
A tangent is perpendicular to the radius at the point of tangency. This forms a 90° angle, a fundamental property of tangent lines.
A tangent is perpendicular to the radius at the point of tangency. This forms a 90° angle, a fundamental property of tangent lines.
← Didn't Know|Knew It →
What condition must hold for a point $P$ to be outside a circle with center $O$ and radius $r$?
What condition must hold for a point $P$ to be outside a circle with center $O$ and radius $r$?
Tap to reveal answer
$OP>r$. Point $P$ must be farther from center $O$ than the radius length.
$OP>r$. Point $P$ must be farther from center $O$ than the radius length.
← Didn't Know|Knew It →
What is the name of the segment from an external point $P$ to the point of tangency $T$ on the circle?
What is the name of the segment from an external point $P$ to the point of tangency $T$ on the circle?
Tap to reveal answer
A tangent segment, $PT$. This segment connects the external point to where the line touches the circle.
A tangent segment, $PT$. This segment connects the external point to where the line touches the circle.
← Didn't Know|Knew It →
What key right triangle is formed when drawing a tangent from external point $P$ to a circle with center $O$?
What key right triangle is formed when drawing a tangent from external point $P$ to a circle with center $O$?
Tap to reveal answer
Right triangle $OPT$ with $OT\perp PT$. The tangent-radius perpendicularity creates this right angle at $T$.
Right triangle $OPT$ with $OT\perp PT$. The tangent-radius perpendicularity creates this right angle at $T$.
← Didn't Know|Knew It →
What is the first construction step after being given circle $(O,r)$ and external point $P$?
What is the first construction step after being given circle $(O,r)$ and external point $P$?
Tap to reveal answer
Draw segment $OP$. This connects the circle center to the external point.
Draw segment $OP$. This connects the circle center to the external point.
← Didn't Know|Knew It →
After drawing $OP$, what point do you construct next to use the diameter method for tangents?
After drawing $OP$, what point do you construct next to use the diameter method for tangents?
Tap to reveal answer
Construct the midpoint $M$ of $OP$. The midpoint will be the center of the auxiliary circle.
Construct the midpoint $M$ of $OP$. The midpoint will be the center of the auxiliary circle.
← Didn't Know|Knew It →
After finding midpoint $M$ of $OP$, what circle do you draw to find tangency points?
After finding midpoint $M$ of $OP$, what circle do you draw to find tangency points?
Tap to reveal answer
Draw the circle centered at $M$ with radius $MO$. This circle has diameter $OP$ and passes through both $O$ and $P$.
Draw the circle centered at $M$ with radius $MO$. This circle has diameter $OP$ and passes through both $O$ and $P$.
← Didn't Know|Knew It →
What are the tangency points in the diameter construction using circle centered at midpoint $M$ of $OP$?
What are the tangency points in the diameter construction using circle centered at midpoint $M$ of $OP$?
Tap to reveal answer
The intersection points $T_1,T_2$ of the two circles. These points lie on both circles, creating right angles at $O$.
The intersection points $T_1,T_2$ of the two circles. These points lie on both circles, creating right angles at $O$.
← Didn't Know|Knew It →
Once tangency points $T_1,T_2$ are found, what lines are the tangents from $P$ to the circle?
Once tangency points $T_1,T_2$ are found, what lines are the tangents from $P$ to the circle?
Tap to reveal answer
Lines $PT_1$ and $PT_2$. These lines through $P$ and the tangency points touch the original circle.
Lines $PT_1$ and $PT_2$. These lines through $P$ and the tangency points touch the original circle.
← Didn't Know|Knew It →
If $OP=r$, how many tangents can be constructed from $P$ to the circle?
If $OP=r$, how many tangents can be constructed from $P$ to the circle?
Tap to reveal answer
$1$ tangent (at $P$). When $P$ is on the circle, only one tangent exists through that point.
$1$ tangent (at $P$). When $P$ is on the circle, only one tangent exists through that point.
← Didn't Know|Knew It →
If $OP<r$, how many real tangents can be constructed from point $P$ to the circle?
If $OP<r$, how many real tangents can be constructed from point $P$ to the circle?
Tap to reveal answer
$0$ real tangents. Points inside the circle cannot have tangent lines to it.
$0$ real tangents. Points inside the circle cannot have tangent lines to it.
← Didn't Know|Knew It →
What is the formula for the length of a tangent segment from $P$ to a circle if $OP=d$ and radius is $r$?
What is the formula for the length of a tangent segment from $P$ to a circle if $OP=d$ and radius is $r$?
Tap to reveal answer
$PT=\sqrt{d^2-r^2}$. Apply Pythagorean theorem to right triangle $OPT$.
$PT=\sqrt{d^2-r^2}$. Apply Pythagorean theorem to right triangle $OPT$.
← Didn't Know|Knew It →
Identify the missing statement: If $PT$ is tangent at $T$, then $\angle OTP=\underline{\quad}$.
Identify the missing statement: If $PT$ is tangent at $T$, then $\angle OTP=\underline{\quad}$.
Tap to reveal answer
$90^\circ$. Tangent-radius perpendicularity creates a right angle.
$90^\circ$. Tangent-radius perpendicularity creates a right angle.
← Didn't Know|Knew It →
What is the formula for tangent length from external point $P$ to a circle when $OP = d$ and radius is $r$?
What is the formula for tangent length from external point $P$ to a circle when $OP = d$ and radius is $r$?
Tap to reveal answer
$PT = \sqrt{d^2 - r^2}$. Derived from Pythagorean theorem on right triangle $OPT$.
$PT = \sqrt{d^2 - r^2}$. Derived from Pythagorean theorem on right triangle $OPT$.
← Didn't Know|Knew It →
What is the key perpendicular relationship between a radius and a tangent at the point of tangency?
What is the key perpendicular relationship between a radius and a tangent at the point of tangency?
Tap to reveal answer
A radius to the point of tangency is perpendicular to the tangent line. This forms a 90° angle, fundamental to tangent-circle relationships.
A radius to the point of tangency is perpendicular to the tangent line. This forms a 90° angle, fundamental to tangent-circle relationships.
← Didn't Know|Knew It →
What condition guarantees that a line is tangent to a circle at point $T$?
What condition guarantees that a line is tangent to a circle at point $T$?
Tap to reveal answer
The line is tangent at $T$ if it is perpendicular to radius $OT$. Forms 90° angle with radius at contact point.
The line is tangent at $T$ if it is perpendicular to radius $OT$. Forms 90° angle with radius at contact point.
← Didn't Know|Knew It →
Which theorem justifies that the constructed points $T$ give right angles in the tangent construction using diameter $OP$?
Which theorem justifies that the constructed points $T$ give right angles in the tangent construction using diameter $OP$?
Tap to reveal answer
Thales' theorem: an angle subtending a diameter is a right angle. Points on semicircle form 90° angles with diameter endpoints.
Thales' theorem: an angle subtending a diameter is a right angle. Points on semicircle form 90° angles with diameter endpoints.
← Didn't Know|Knew It →
How many tangents can be drawn from a point $P$ outside a circle (in the Euclidean plane)?
How many tangents can be drawn from a point $P$ outside a circle (in the Euclidean plane)?
Tap to reveal answer
Two tangents. One on each side of line $OP$ through the center.
Two tangents. One on each side of line $OP$ through the center.
← Didn't Know|Knew It →
What is the length relationship between the two tangent segments from the same external point $P$ to a circle?
What is the length relationship between the two tangent segments from the same external point $P$ to a circle?
Tap to reveal answer
They are congruent: $PT_1 = PT_2$. Both tangent segments from external point have equal length.
They are congruent: $PT_1 = PT_2$. Both tangent segments from external point have equal length.
← Didn't Know|Knew It →
Identify the necessary and sufficient condition for tangents from $P$ to exist using distance $OP$ and radius $r$.
Identify the necessary and sufficient condition for tangents from $P$ to exist using distance $OP$ and radius $r$.
Tap to reveal answer
Tangents exist iff $OP > r$. Point must be outside circle for tangents to exist.
Tangents exist iff $OP > r$. Point must be outside circle for tangents to exist.
← Didn't Know|Knew It →
What is the value of $PT$ if a circle has radius $r = 5$ and $OP = 13$?
What is the value of $PT$ if a circle has radius $r = 5$ and $OP = 13$?
Tap to reveal answer
$PT = 12$. Using $PT = \sqrt{13^2 - 5^2} = \sqrt{169-25} = \sqrt{144}$.
$PT = 12$. Using $PT = \sqrt{13^2 - 5^2} = \sqrt{169-25} = \sqrt{144}$.
← Didn't Know|Knew It →
What is the value of $PT$ if a circle has radius $r = 6$ and $OP = 10$?
What is the value of $PT$ if a circle has radius $r = 6$ and $OP = 10$?
Tap to reveal answer
$PT = 8$. Using $PT = \sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64}$.
$PT = 8$. Using $PT = \sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64}$.
← Didn't Know|Knew It →
What is the value of $OP$ if $r = 9$ and tangent length $PT = 12$ from external point $P$?
What is the value of $OP$ if $r = 9$ and tangent length $PT = 12$ from external point $P$?
Tap to reveal answer
$OP = 15$. From $12^2 + 9^2 = OP^2$, so $144 + 81 = 225$.
$OP = 15$. From $12^2 + 9^2 = OP^2$, so $144 + 81 = 225$.
← Didn't Know|Knew It →
What is the equation of the tangent line to $x^2 + y^2 = r^2$ at point $(x_1, y_1)$ on the circle?
What is the equation of the tangent line to $x^2 + y^2 = r^2$ at point $(x_1, y_1)$ on the circle?
Tap to reveal answer
$x x_1 + y y_1 = r^2$. Point-slope form using gradient perpendicular to radius.
$x x_1 + y y_1 = r^2$. Point-slope form using gradient perpendicular to radius.
← Didn't Know|Knew It →