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Circle Geometry: Angles and Segments

2 questions

Question 1Question →

Chords ABAB and CDCD intersect at point EE inside a circle. If AE=6AE = 6, EB=8EB = 8, and the total length of chord CDCD is 1616, what is the length of the shorter segment of chord CDCD?

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Answer: 4

Answer

The length of the shorter segment of chord CDCD is 44.
According to the Intersecting Chords Theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other. For chords ABAB and CDCD intersecting at point EE, this relationship is expressed as AE⋅EB=CE⋅EDAE \cdot EB = CE \cdot ED. Substituting the given values yields 6⋅8=CE⋅ED6 \cdot 8 = CE \cdot ED, so CE⋅ED=48CE \cdot ED = 48. Since the total length of chord CDCD is 1616, we can define CE=xCE = x and ED=16−xED = 16 - x. The equation becomes x(16−x)=48x(16 - x) = 48, which simplifies to the quadratic equation x2−16x+48=0x^2 - 16x + 48 = 0. Factoring this equation gives (x−12)(x−4)=0(x - 12)(x - 4) = 0, meaning the two segments of chord CDCD have lengths of 1212 and 44. The length of the shorter segment is 44.

Step-by-Step Solution

1
State the relationship between intersecting chord segments.
AE⋅EB=CE⋅EDAE \cdot EB = CE \cdot ED
By the Intersecting Chords Theorem, the product of the segments of one chord equals the product of the segments of the other.
2
Substitute the known lengths and define the segments of CDCD using a variable xx.
6⋅8=x(16−x)6 \cdot 8 = x(16 - x), which simplifies to 48=16x−x248 = 16x - x^2.
We are given AE=6AE = 6 and EB=8EB = 8. Since the total length of chord CDCD is 1616, if one segment is xx, the remaining segment must be 16−x16 - x.
3
Solve the quadratic equation for xx by factoring.
x2−16x+48=0  ⟹  (x−12)(x−4)=0x^2 - 16x + 48 = 0 \implies (x - 12)(x - 4) = 0, so x=12x = 12 or x=4x = 4.
Rearranging the equation into standard quadratic form allows us to find the two possible segment lengths.
4
Identify the shorter segment length from the two solutions.
44
The two segment lengths are 1212 and 44. The problem asks for the shorter segment, which is 44.

Key Concept

Intersecting Chords Theorem
Question 2Question →

A tangent line segment PT‾\overline{PT} touches a circle at point TT. A secant line from external point PP intersects the circle at points AA and BB, such that point AA lies on segment PB‾\overline{PB}. If m∠P=35∘m\angle P = 35^\circ and the measure of minor arc ATAT is 50∘50^\circ, what is the degree measure of inscribed angle ∠TAB\angle TAB?

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Answer: 60

Answer

The degree measure of inscribed angle ∠TAB\angle TAB is 60 degrees.
According to the exterior angle theorem for circles, the angle formed by a tangent and a secant meeting at an external point PP is equal to half the difference of the intercepted arcs: m∠P=12(mBT^−mAT^)m\angle P = \frac{1}{2}(m\widehat{BT} - m\widehat{AT}). Substituting m∠P=35∘m\angle P = 35^\circ and mAT^=50∘m\widehat{AT} = 50^\circ into the equation gives 35∘=12(mBT^−50∘)35^\circ = \frac{1}{2}(m\widehat{BT} - 50^\circ), which simplifies to mBT^=120∘m\widehat{BT} = 120^\circ. The inscribed angle ∠TAB\angle TAB intercepts arc BTBT. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc, giving m∠TAB=12(120∘)=60∘m\angle TAB = \frac{1}{2}(120^\circ) = 60^\circ. Alternatively, inside triangle PATPAT, the tangent-chord angle PTAPTA intercepts arc ATAT, so m∠PTA=12(50∘)=25∘m\angle PTA = \frac{1}{2}(50^\circ) = 25^\circ. Since the angles in triangle PATPAT sum to 180∘180^\circ, m∠PAT=180∘−(35∘+25∘)=120∘m\angle PAT = 180^\circ - (35^\circ + 25^\circ) = 120^\circ. Angle TABTAB is supplementary to angle PATPAT, so m∠TAB=180∘−120∘=60∘m\angle TAB = 180^\circ - 120^\circ = 60^\circ.

Step-by-Step Solution

1
Use the exterior angle relationship for the secant and tangent to find the measure of arc BTBT.
mBT^=120∘m\widehat{BT} = 120^\circ
The exterior angle measure equals half the difference of intercepted arcs BTBT and ATAT: 35∘=12(mBT^−50∘)35^\circ = \frac{1}{2}(m\widehat{BT} - 50^\circ).
2
Use the Inscribed Angle Theorem to find m∠TABm\angle TAB.
m∠TAB=60∘m\angle TAB = 60^\circ
An inscribed angle measure is equal to half the measure of its intercepted arc: m∠TAB=12(120∘)=60∘m\angle TAB = \frac{1}{2}(120^\circ) = 60^\circ.

Key Concept

Secant-Tangent Angle Theorem and Inscribed Angle Theorem
Estimated Time:1m 30s