14.1 Circle Angles, Chords, Tangents, Power of a Point (Kuvvet) & Sector/Annulus Area

14.1 Circle Angles, Chords, Tangents, Power of a Point (Kuvvet) & Sector/Annulus Area

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1. Circle Angles (Çemberde Açılar)

A full circle measures $360^\circ$. Every angle associated with a circle is determined by the degree measure of the arc(s) it intercepts.

Central, Inscribed, and Tangent-Chord Angles

1. Central Angle (Merkez Açı): Has its vertex at the center $O$ of the circle. Its measure is equal to the measure of its intercepted arc:

$$\alpha = m(\angle AOB) = m(\widehat{AB})$$

1. Inscribed Angle (Çevre Açı): Has its vertex on the circumference and sides that are chords. Its measure is half the measure of its intercepted arc:

$$\alpha = m(\angle ACB) = \frac{1}{2}m(\widehat{AB})$$

Consequence: All inscribed angles that subtend the same arc are equal, and an inscribed angle is half the central angle subtending the same arc. 3. Tangent-Chord Angle (Teğet-Kiriş Açı): Formed by a tangent line and a chord meeting at the point of tangency. Like an inscribed angle, it equals half the arc enclosed by the chord:

$$\alpha = \frac{1}{2}m(\widehat{AB})$$

1. Thales' Circle Theorem (Çapı Gören Çevre Açı): Because a semicircle measures $180^\circ$, any inscribed angle that subtends a diameter is a right angle ($90^\circ$).

Interior, Exterior, and Two-Tangent Angles

• Interior Secant Angle (İç Açı): When two chords $AC$ and $BD$ intersect inside the circle at point $P$, the vertical angles equal the average of the two opposite intercepted arcs $x = m(\widehat{AB})$ and $y = m(\widehat{CD})$:

$$\alpha = \frac{x + y}{2}$$

• Exterior Secant/Tangent Angle (Dış Açı): When two secants, a secant and a tangent, or two tangents intersect at an exterior point $P$, the angle equals half the difference between the far arc $x$ and the near arc $y$:

$$\alpha = \frac{x - y}{2}$$

• Two-Tangent Hat Rule (Külah / Dondurma Külahı Kuralı): When both rays from an exterior point $P$ are tangent to the circle at $A$ and $B$, the exterior angle $\angle APB$ and the minor arc $\widehat{AB}$ are supplementary:

$$m(\angle APB) + m(\widehat{AB}) = 180^\circ$$

Furthermore, the line $PO$ joining $P$ to the center $O$ bisects both the angle $\angle APB$ and the arc $\widehat{AB}$, creating a half-hat where:

$$m(\angle APO) + m(\widehat{AC}) = 90^\circ$$

Chord-Arc Symmetry & Cyclic Quadrilaterals (Kirişler Dörtgeni)

• Equal Chords (Eş Kirişler): Congruent chords cut off equal arcs: $|AB| = |CD| \iff m(\widehat{AB}) = m(\widehat{CD})$.
• Parallel Chords (Paralel Kirişler): Two parallel chords cut off equal arcs between them: $AB \parallel CD$ with $A,B,C,D$ in cyclic order implies $m(\widehat{AD})=m(\widehat{BC})$ for the intervening arcs.
• Cyclic Quadrilateral (Kirişler Dörtgeni): A quadrilateral $ABCD$ whose four vertices lie on a circle has opposite interior angles that sum to $180^\circ$:

$$m(\angle A) + m(\angle C) = 180^\circ \quad \text{and} \quad m(\angle B) + m(\angle D) = 180^\circ$$

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2. Chords, Tangents & Power of a Point (Çemberde Uzunluk ve Kuvvet)

Fundamental Center-to-Chord & Center-to-Tangent Rules

1. Perpendicular from Center to a Chord: A perpendicular segment $OH$ dropped from the center $O$ to a chord $AB$ bisects the chord ($|AH| = |HB| = \frac{|AB|}{2}$) and bisects the subtended arc. With radius $R$ and distance $d = |OH|$, $\triangle OHB$ is a right triangle satisfying:

$$R^2 = d^2 + \left(\frac{|AB|}{2}\right)^2$$

Chords closer to the center are longer: $d_1 < d_2 \iff |AB| > |CD|$. 2. Radius-to-Tangent Perpendicularity: The radius $OT$ drawn to a point of tangency $T$ is always perpendicular to the tangent line ($OT \perp d$, forming a $90^\circ$ angle). 3. Two Tangents & Tangential Quadrilaterals (Teğetler Dörtgeni): Two tangent segments drawn from the same external point $P$ have equal length ($|PA| = |PB|$). Consequently, if a quadrilateral $ABCD$ circumscribes an inscribed circle (so all four sides are tangent to the circle), the sums of its opposite side lengths are equal:

$$a + c = b + d$$

And its area equals $A = u \cdot r$, where $u = \frac{a+b+c+d}{2} = a + c$ is the semiperimeter and $r$ is the inradius.

Power of a Point (Çemberde Kuvvet)

For any point $P$ at distance $|OP|$ from the center of a circle of radius $R$, the product of directed segment lengths along any line through $P$ intersecting the circle is constant:

• Intersecting Chords / Interior Power (İç Kuvvet): If chords $AB$ and $CD$ intersect at an interior point $P$:

$$|PA| \cdot |PB| = |PC| \cdot |PD| = R^2 - |OP|^2$$

• Two Exterior Secants (Dış Kuvvet): If two secants from an exterior point $P$ cut the circle at $A, B$ and $C, D$ respectively, multiply near segment $\times$ whole secant:

$$|PA| \cdot |PB| = |PC| \cdot |PD|$$

• Tangent-Secant Theorem (Teğet-Kesen Kuvveti): If a tangent $PT$ and a secant $PAB$ are drawn from an exterior point $P$:

$$|PT|^2 = |PA| \cdot |PB| = |OP|^2 - R^2$$

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3. Circle Circumference, Sector Area & Annulus (Dairede Alan ve Çevre)

For a circle of radius $R$ and a sector with central angle $\alpha$ (in degrees):

• Circumference (Çevre): $C = 2\pi R$, and Arc Length (Yay Uzunluğu): $|\widehat{AB}| = 2\pi R \cdot \frac{\alpha}{360^\circ}$.
• Full Disk Area (Dairenin Alanı): $A = \pi R^2$.
• Sector Area (Daire Diliminin Alanı):

$$A_{\text{sector}} = \pi R^2 \cdot \frac{\alpha}{360^\circ} = \frac{1}{2} \cdot |\widehat{AB}| \cdot R$$

Notice that $\frac{1}{2} \cdot |\widehat{AB}| \cdot R$ treats the sector like a triangle with base $|\widehat{AB}|$ and height $R$.

• Circular Segment Area (Daire Parçası / Kesmesi Alanı): The region bounded by chord $AB$ and arc $\widehat{AB}$ equals the sector area minus the area of isosceles triangle $\triangle OAB$:

$$A_{\text{segment}} = \pi R^2 \cdot \frac{\alpha}{360^\circ} - \frac{1}{2}R^2\sin\alpha$$

• Annulus Area (Halka Alanı): Between two concentric circles of radii $R > r$, the ring area is $A_{\text{annulus}} = \pi(R^2 - r^2)$. If a chord $AB$ of the outer circle is tangent to the inner circle at $T$, then $R^2 - r^2 = |AT|^2 = \left(\frac{|AB|}{2}\right)^2$, yielding the shortcut:

$$A_{\text{annulus}} = \pi \left(\frac{|AB|}{2}\right)^2$$

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Summary Reference Table: Circle Rules

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Worked Examples

Worked Example 1: Chord Bisection and Tangent-Secant Power

Problem: A circle with center $O$ has radius $R = 10\text{ cm}$, and a chord $AB$ has length $|AB| = 16\text{ cm}$. Point $P$ lies outside the circle on the ray $AB$ extended beyond $B$ such that $|PB| = 4\text{ cm}$, and $PT$ is a tangent segment from $P$ to the circle. Find:

1. The perpendicular distance $d = |OH|$ from the center $O$ to the chord $AB$.
2. The length of the tangent segment $|PT|$.

Step-by-Step Solution:

1. Drop a perpendicular to bisect the chord: Drawing $OH \perp AB$ divides the chord into two equal halves:

$$|AH| = |HB| = \frac{16}{2} = 8\text{ cm}$$

In right triangle $\triangle OHB$ with hypotenuse $R = 10\text{ cm}$, apply the Pythagorean theorem:

$$d = |OH| = \sqrt{R^2 - |HB|^2} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = 6\text{ cm}$$

1. Apply the Tangent-Secant Power Theorem: From external point $P$, the near segment is $|PB| = 4\text{ cm}$ and the whole secant is $|PA| = |PB| + |BA| = 4 + 16 = 20\text{ cm}$. Therefore:

$$|PT|^2 = |PB| \cdot |PA| = 4 \cdot 20 = 80 \implies |PT| = \sqrt{80} = 4\sqrt{5}\text{ cm}$$

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Worked Example 2: Two-Tangent Hat Rule and Circular Segment Area

Problem: From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with center $O$ and radius $R = 6\text{ cm}$ such that $m(\angle APB) = 60^\circ$. Find:

1. The degree measure of the minor arc $m(\widehat{AB})$.
2. The exact area of the circular segment (daire parçası) bounded by chord $AB$ and minor arc $\widehat{AB}$.

Step-by-Step Solution:

1. Find the arc and central angle: By the Two-Tangent Hat Rule (Külah Kuralı), the exterior angle and minor arc sum to $180^\circ$:

$$m(\widehat{AB}) = 180^\circ - m(\angle APB) = 180^\circ - 60^\circ = 120^\circ$$

Hence, the central angle is $\alpha = m(\angle AOB) = 120^\circ$. 2. Compute the circular segment area: Subtract the area of $\triangle OAB$ from the area of sector $OAB$:

$$A_{\text{sector}} = \pi R^2 \cdot \frac{120^\circ}{360^\circ} = \pi (6^2) \cdot \frac{1}{3} = 12\pi\text{ cm}^2$$ $$A_{\triangle OAB} = \frac{1}{2} R^2 \sin 120^\circ = \frac{1}{2}(36)\left(\frac{\sqrt{3}}{2}\right) = 9\sqrt{3}\text{ cm}^2$$ $$A_{\text{segment}} = 12\pi - 9\sqrt{3}\text{ cm}^2$$

When using the half-hat formula, $C$ denotes the midpoint of the minor arc AB on the line PO. Equal-chord comparisons use matching minor arcs in the same circle. Signed power is $OP^2-R^2$; the interior unsigned chord-product formula above uses its positive magnitude.

Review the Method

A circular sector of arc length |arc AB| and radius R has area (1/2) * |arc AB| * R, and an annulus (halka) whose inner circle is tangent to an outer chord AB has area pi * (|AB|/2)^2