Circle Relationships: Angles, Radii, and Chords
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Geometry › Circle Relationships: Angles, Radii, and Chords
An inscribed angle and a central angle both intercept the same arc of a circle. If the central angle measures $$3x + 20°$$ and the inscribed angle measures $$2x - 5°$$, what is the value of $$x$$?
15
25
30
35
Explanation
An inscribed angle measures half of the central angle that intercepts the same arc. Therefore: $$2x - 5 = \frac{1}{2}(3x + 20)$$. Solving: $$2x - 5 = \frac{3x + 20}{2}$$, so $$2(2x - 5) = 3x + 20$$, giving $$4x - 10 = 3x + 20$$, and $$x = 30$$. Choice A (15) might result from setting the angles equal: $$2x - 5 = 3x + 20$$. Choice B (25) could come from solving $$2x = 3x + 20 - 5$$. Choice D (35) might result from incorrectly manipulating the equation.
Two chords $$AB$$ and $$CD$$ intersect inside circle $$O$$ at point $$E$$. If $$AE = 6$$, $$EB = 4$$, and $$CE = 8$$, what is the length of $$ED$$?
2
3
5
12
Explanation
When two chords intersect inside a circle, the products of their segments are equal: $$AE \cdot EB = CE \cdot ED$$. Substituting: $$6 \cdot 4 = 8 \cdot ED$$, so $$24 = 8 \cdot ED$$, giving $$ED = 3$$. Choice B (2) might result from using $$AE - EB = ED$$. Choice C (5) could come from incorrectly using $$\frac{AE + EB}{2} - 1 = ED$$. Choice D (12) might result from using $$CE + EB = ED$$.
In circle $$O$$, tangent $$PT$$ touches the circle at point $$T$$, and secant $$PAB$$ passes through the circle with $$A$$ and $$B$$ on the circle. If $$PT = 12$$ and $$PA = 8$$, what is the length of $$PB$$?
16
18
20
24
Explanation
By the tangent-secant theorem, when a tangent and secant are drawn from the same external point, $$PT^2 = PA \cdot PB$$. Substituting the known values: $$12^2 = 8 \cdot PB$$, so $$144 = 8 \cdot PB$$, giving $$PB = 18$$. Choice A (16) might result from incorrectly using $$PT \cdot PA = PB$$. Choice C (20) could come from using $$PT + PA = PB$$. Choice D (24) might result from using $$2 \cdot PT = PB$$.
In circle $$M$$, radius $$MR$$ is perpendicular to chord $$ST$$ at point $$H$$. If the radius of the circle is 13 and $$MH = 5$$, what is the length of chord $$ST$$?
12
18
24
26
Explanation
When a radius is perpendicular to a chord, it bisects the chord. In right triangle $$MHS$$ (where $$S$$ is one endpoint of the chord), we have $$MS = 13$$ (radius), $$MH = 5$$, and $$HS$$ can be found using the Pythagorean theorem: $$HS^2 + MH^2 = MS^2$$, so $$HS^2 + 25 = 169$$, giving $$HS^2 = 144$$ and $$HS = 12$$. Since the radius bisects the chord, $$ST = 2 \cdot HS = 2 \cdot 12 = 24$$. Choice A (12) is just half the chord length. Choice B (18) might result from adding $$MH + MS = 5 + 13 = 18$$. Choice D (26) might come from adding $$2 \cdot MS = 2 \cdot 13 = 26$$.
In circle $$P$$, two secants are drawn from external point $$R$$. Secant $$RST$$ has $$RS = 9$$ and $$ST = 7$$. Secant $$RUV$$ has $$RU = 6$$ and $$UV = x$$. What is the value of $$x$$?
8
12
18
21
Explanation
By the secant-secant theorem, when two secants are drawn from the same external point: $$RS \cdot RT = RU \cdot RV$$, where $$RT = RS + ST$$ and $$RV = RU + UV$$. So: $$RT = 9 + 7 = 16$$ and $$RV = 6 + x$$. The equation becomes: $$9 \cdot 16 = 6 \cdot(6 + x)$$, so $$144 = 6(6 + x) = 36 + 6x$$, giving $$6x = 108$$ and $$x = 18$$. Choice A (8) might result from using $$RS - RU + ST = x$$. Choice B (12) could come from incorrectly using $$\frac{RS \cdot ST}{RU} = x$$. Choice D (21) might result from using $$RS + ST + RU - 1 = x$$.