Constructing Tangents to Circles - Precalculus

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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