Precalculus Flashcards: Constructing Tangents To Circles

What this deck covers

This deck focuses on Constructing Tangents To Circles, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the tangent length: $r=6$ and $OP=10$; what is $PT$?

Answer: $8$. Use $PT=\sqrt{10^2-6^2}=\sqrt{100-36}=\sqrt{64}$.

Flashcard 2: What condition guarantees that a line is tangent to a circle at point $T$?

Answer: The line is tangent at $T$ if it is perpendicular to radius $OT$. Forms 90° angle with radius at contact point.

Flashcard 3: What is the tangent line to $(x-1)^2 + (y+2)^2 = 25$ at $(5,1)$?

Answer: $4(x-1) + 3(y+2) = 25$. Apply formula with center $(1,-2)$ and point $(5,1)$.

Flashcard 4: What theorem relates a tangent line to the radius at the point of tangency?

Answer: A tangent is perpendicular to the radius at the point of tangency. This forms a 90° angle, a fundamental property of tangent lines.

Flashcard 5: Identify the missing statement: If $PT$ is tangent at $T$, then $\angle OTP=\underline{\quad}$.

Answer: $90^\circ$. Tangent-radius perpendicularity creates a right angle.

Flashcard 6: If $OP=r$, how many tangents can be constructed from $P$ to the circle?

Answer: $1$ tangent (at $P$). When $P$ is on the circle, only one tangent exists through that point.

Flashcard 7: Find the tangent length: a circle has $r=5$ and an external point has $OP=13$; what is $PT$?

Answer: $12$. Use $PT=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}$.

Flashcard 8: After drawing $OP$, what point do you construct next to use the diameter method for tangents?

Answer: Construct the midpoint $M$ of $OP$. The midpoint will be the center of the auxiliary circle.

Flashcard 9: Identify the circle property used to check a constructed tangent: what must be true about $OT$ and $PT$ at tangency point $T$?

Answer: $OT \perp PT$ at $T$. Radius and tangent must be perpendicular at contact point.

Flashcard 10: What is the tangent line to $x^2 + y^2 = 16$ at $(0,4)$?

Answer: $y = 4$. Horizontal tangent at top of circle (vertical radius).

Flashcard 11: What fact about tangent segments is true if two tangents are drawn from the same external point $P$?

Answer: $PT_1=PT_2$. Both tangent segments from the same external point have equal length.

Flashcard 12: What condition must hold for a point $P$ to be outside a circle with center $O$ and radius $r$?

Answer: $OP>r$. Point $P$ must be farther from center $O$ than the radius length.

Flashcard 13: What is the length relationship between the two tangent segments from the same external point $P$ to a circle?

Answer: They are congruent: $PT_1 = PT_2$. Both tangent segments from external point have equal length.

Flashcard 14: What is the slope of the tangent to $x^2 + y^2 = r^2$ at $(x_1, y_1)$ when $y_1 \ne 0$?

Answer: $m_{\text{tan}} = -\frac{x_1}{y_1}$. Perpendicular to radius slope $\frac{y_1}{x_1}$.

Flashcard 15: Once tangency points $T_1,T_2$ are found, what lines are the tangents from $P$ to the circle?

Answer: Lines $PT_1$ and $PT_2$. These lines through $P$ and the tangency points touch the original circle.

Flashcard 16: What is the tangent line to $x^2 + y^2 = 9$ at $(3,0)$?

Answer: $x = 3$. Vertical tangent at rightmost point (horizontal radius).

Flashcard 17: Which theorem justifies that the constructed points $T$ give right angles in the tangent construction using diameter $OP$?

Answer: Thales' theorem: an angle subtending a diameter is a right angle. Points on semicircle form 90° angles with diameter endpoints.

Flashcard 18: What is the key perpendicular relationship between a radius and a tangent at the point of tangency?

Answer: A radius to the point of tangency is perpendicular to the tangent line. This forms a 90° angle, fundamental to tangent-circle relationships.

Flashcard 19: What key right triangle is formed when drawing a tangent from external point $P$ to a circle with center $O$?

Answer: Right triangle $OPT$ with $OT\perp PT$. The tangent-radius perpendicularity creates this right angle at $T$.

Flashcard 20: What is the tangent line to $x^2 + y^2 = 25$ at $(3,4)$?

Answer: $3x + 4y = 25$. Substituting $(3,4)$ into $xx_1 + yy_1 = r^2$ with $r^2=25$.

Flashcard 21: Identify the number of tangents: if $OP=7$ and $r=7$, how many tangents from $P$ exist?

Answer: $1$. When $OP=r$, point $P$ is on the circle with one tangent.

Flashcard 22: What are the tangency points in the diameter construction using circle centered at midpoint $M$ of $OP$?

Answer: The intersection points $T_1,T_2$ of the two circles. These points lie on both circles, creating right angles at $O$.

Flashcard 23: Identify the number of tangents: if $OP=9$ and $r=4$, how many tangents from $P$ exist?

Answer: $2$. Since $OP>r$, point $P$ is external, allowing two tangents.

Flashcard 24: What is the value of $PT$ if a circle has radius $r = 5$ and $OP = 13$?

Answer: $PT = 12$. Using $PT = \sqrt{13^2 - 5^2} = \sqrt{169-25} = \sqrt{144}$.

Flashcard 25: What is the value of $PT$ if a circle has radius $r = 6$ and $OP = 10$?

Answer: $PT = 8$. Using $PT = \sqrt{10^2 - 6^2} = \sqrt{100-36} = \sqrt{64}$.

Flashcard 26: What is the name of the segment from an external point $P$ to the point of tangency $T$ on the circle?

Answer: A tangent segment, $PT$. This segment connects the external point to where the line touches the circle.

Flashcard 27: If $OP<r$, how many real tangents can be constructed from point $P$ to the circle?

Answer: $0$ real tangents. Points inside the circle cannot have tangent lines to it.

Flashcard 28: What is the value of $OP$ if $r = 9$ and tangent length $PT = 12$ from external point $P$?

Answer: $OP = 15$. From $12^2 + 9^2 = OP^2$, so $144 + 81 = 225$.

Flashcard 29: What is the first construction step after being given circle $(O,r)$ and external point $P$?

Answer: Draw segment $OP$. This connects the circle center to the external point.

Flashcard 30: What is the equation of the tangent line to $(x-h)^2 + (y-k)^2 = r^2$ at point $(x_1, y_1)$ on the circle?

Answer: $(x_1-h)(x-h) + (y_1-k)(y-k) = r^2$. Translates standard form by center coordinates $(h,k)$.

Flashcard 31: Identify the number of tangents: if $OP=3$ and $r=5$, how many real tangents from $P$ exist?

Answer: $0$. Since $OP<r$, point $P$ is inside the circle with no tangents.

Flashcard 32: After finding midpoint $M$ of $OP$, what circle do you draw to find tangency points?

Answer: Draw the circle centered at $M$ with radius $MO$. This circle has diameter $OP$ and passes through both $O$ and $P$.

Flashcard 33: Identify the necessary and sufficient condition for tangents from $P$ to exist using distance $OP$ and radius $r$.

Answer: Tangents exist iff $OP > r$. Point must be outside circle for tangents to exist.

Flashcard 34: What is the formula for tangent length from external point $P$ to a circle when $OP = d$ and radius is $r$?

Answer: $PT = \sqrt{d^2 - r^2}$. Derived from Pythagorean theorem on right triangle $OPT$.

Flashcard 35: How many tangents can be drawn from a point $P$ outside a circle (in the Euclidean plane)?

Answer: Two tangents. One on each side of line $OP$ through the center.

Flashcard 36: What is the equation of the tangent line to $x^2 + y^2 = r^2$ at point $(x_1, y_1)$ on the circle?

Answer: $x x_1 + y y_1 = r^2$. Point-slope form using gradient perpendicular to radius.

Flashcard 37: What is the formula for the length of a tangent segment from $P$ to a circle if $OP=d$ and radius is $r$?

Answer: $PT=\sqrt{d^2-r^2}$. Apply Pythagorean theorem to right triangle $OPT$.