Math 3 Quiz: Chord Secant And Tangent Relationships
11 questions · exam conditions
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Chord Secant And Tangent RelationshipsQuestion 1 of 11

From point A outside circle O, a tangent and a secant are drawn. The tangent touches the circle at B, and the secant passes through points C and D on the circle (C is between A and D). If AB = 2212\sqrt{21} and AC = 7, what is the length of AD?

12
14
16
21
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Math 3 Quiz

Math 3 Quiz: Chord Secant And Tangent Relationships

Practice Chord Secant And Tangent Relationships in Math 3 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Chord Secant And Tangent Relationships, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 3.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

From point A outside circle O, a tangent and a secant are drawn. The tangent touches the circle at B, and the secant passes through points C and D on the circle (C is between A and D). If AB = 2212\sqrt{21} and AC = 7, what is the length of AD?

  1. 12 (correct answer)
  2. 14
  3. 16
  4. 21
Explanation: Using the tangent-secant theorem: AB² = AC × AD. We have (2√21)² = 7 × AD, so 4 × 21 = 7 × AD, giving 84 = 7 × AD, therefore AD = 12. Choice B incorrectly adds AB and AC. Choice C uses an incorrect doubling of AB. Choice D uses AB² ÷ 4 instead of the correct relationship.

Question 2

Two secants are drawn from external point W to circle O. One secant intersects the circle at X and Y (X between W and Y), and the other intersects at points Z and V (Z between W and V). If WX = 6, XY = 10, and WZ = 8, what is WV?

  1. 12 (correct answer)
  2. 15
  3. 18
  4. 20
Explanation: For two secants from an external point: WX × WY = WZ × WV. First, WY = WX + XY = 6 + 10 = 16. Then 6 × 16 = 8 × WV, so 96 = 8 × WV, giving WV = 12. Choice B incorrectly uses WV = WX + XY - 1. Choice C uses WV = WY + 2. Choice D assumes WV = WX + XY + WZ - 2.

Question 3

From external point T, two tangent segments are drawn to circle O, touching at points U and V. A secant from T intersects the circle at points W and X (W between T and X). If TU = 20 and TW = 16, what is TX?

  1. 24
  2. 25 (correct answer)
  3. 30
  4. 36
Explanation: Since tangent segments from an external point are equal, TV = TU = 20. Using the tangent-secant theorem: TU² = TW × TX. So 20² = 16 × TX, giving 400 = 16 × TX, therefore TX = 25. Choice A incorrectly uses TX = TU + 4. Choice C uses TX = TW + TU - 6. Choice D assumes TX = TU + TW incorrectly.

Question 4

Point W lies outside circle V. A secant from W passes through points A and B on the circle (A closer to W), with WA = 8 and AB = 10. Another line from W is tangent to the circle at point C. If a third secant from W passes through points D and E (D closer to W) with WD = 6, what is the length DE?

  1. 18 (correct answer)
  2. 20
  3. 22
  4. 24
Explanation: Use the secant-secant theorem with the two secants: WA × WB = WD × WE. We have WB = WA + AB = 8 + 10 = 18. So 8 × 18 = 6 × WE, giving 144 = 6 × WE, thus WE = 24. Therefore DE = WE - WD = 24 - 6 = 18. Choice B might result from adding WA + AB incorrectly. Choice C could come from miscalculation in the proportion. Choice D incorrectly uses WE as the answer.

Question 5

In circle O, two chords AB and CD intersect at point P inside the circle. If AP = 6, PB = 4, and CP = 8, what is the length of PD?

  1. 3 (correct answer)
  2. 4
  3. 5
  4. 6
Explanation: When two chords intersect inside a circle, the products of their segments are equal: AP × PB = CP × PD. Substituting: 6 × 4 = 8 × PD, so 24 = 8 × PD, therefore PD = 3. Choice B uses PB incorrectly. Choice C assumes the segments form a 3-4-5 pattern. Choice D incorrectly uses AP as the answer.

Question 6

From an external point Q, two secants are drawn to circle R. One secant passes through points A and B (with A closer to Q), and the other passes through points C and D (with C closer to Q). If QA = 5, AB = 7, and QC = 4, what is the length of CD?

  1. 8
  2. 9 (correct answer)
  3. 11
  4. 12
Explanation: For two secants from an external point, QA × QB = QC × QD. First, QB = QA + AB = 5 + 7 = 12. Then 5 × 12 = 4 × QD, so QD = 15. Therefore CD = QD - QC = 15 - 4 = 9. Choice A uses AB incorrectly. Choice C adds QA and AB incorrectly. Choice D uses QB as the answer.

Question 7

From point M outside circle N, a tangent MT is drawn touching the circle at T, and a secant is drawn through points X and Y on the circle (X closer to M). If MT = 15 and the secant has total length MY = 25, what is the length MX?

  1. 9 (correct answer)
  2. 10
  3. 15
  4. 16
Explanation: Using the tangent-secant theorem: MT² = MX × MY. So 15² = MX × 25, giving 225 = 25 × MX, therefore MX = 9. Choice B would result from a calculation error. Choice C incorrectly uses the tangent length. Choice D comes from incorrectly using 16 × 25 = 400.

Question 8

Two chords of circle O intersect at point K inside the circle. One chord has segments of length a and b on either side of K, while the other has segments of length c and d. If a:b = 2:3 and c:d = 4:1, and the product ab = 24, what is the value of cd?

  1. 20
  2. 24 (correct answer)
  3. 28
  4. 32
Explanation: Since the chords intersect inside the circle, ab = cd. Given ab = 24, we have cd = 24. The ratios a:b = 2:3 and c:d = 4:1 are additional information that confirms our segments are valid, but the key theorem is that the products must be equal. Choice A might come from incorrectly using the ratio 4:1 to get 4×5=20. Choice C could result from adding the ratio parts. Choice D might come from using 2×16=32.

Question 9

A tangent from external point T touches circle O at point A, and a secant from T passes through points B and C on the circle (with B between T and C). If TA = 12 and TB = 8, what is the length of TC?

  1. 16
  2. 18 (correct answer)
  3. 20
  4. 24
Explanation: For a tangent and secant from the same external point, TA² = TB × TC. So 12² = 8 × TC, which gives 144 = 8 × TC, therefore TC = 18. Choice A incorrectly uses TA = TB + BC. Choice C assumes TC = TB + TA incorrectly. Choice D uses 2 × TA instead of TA².

Question 10

In circle O, two chords AB and CD intersect at point P inside the circle. If AP = 6, PB = 8, and CP = 4, what is the length of PD?

  1. 14
  2. 10
  3. 12 (correct answer)
  4. 16
Explanation: When you see two chords intersecting inside a circle, you're dealing with the intersecting chords theorem. This powerful geometric principle states that when two chords intersect at a point inside a circle, the products of their segments are equal. For chords AB and CD intersecting at point P, the theorem tells us that AP×PB=CP×PDAP \times PB = CP \times PD. This relationship holds because the angles formed create similar triangles, making the cross-products equal. Let's apply this to our problem. We know AP = 6, PB = 8, and CP = 4, and we need to find PD. Setting up the equation: 6×8=4×PD6 \times 8 = 4 \times PD 48=4×PD48 = 4 \times PD PD=12PD = 12 Looking at the wrong answers: Choice A (14) might result from incorrectly adding AP + PB rather than multiplying them. Choice B (10) could come from mistakenly calculating AP×PBCP=484−2=10\frac{AP \times PB}{CP} = \frac{48}{4} - 2 = 10 through some algebraic error. Choice D (16) might arise from incorrectly thinking the relationship is AP+PB=CP+PDAP + PB = CP + PD, which would give 6+8=4+PD6 + 8 = 4 + PD, so PD=10PD = 10, but then doubling due to confusion. Study tip: Memorize the intersecting chords theorem as "segment times segment equals segment times segment." Whenever you see chords crossing inside a circle with three given lengths, immediately set up the cross-multiplication equation. This pattern appears frequently in geometry problems.

Question 11

A tangent from point T touches circle M at point N, and a secant from T passes through points P and Q on the circle (with P closer to T). If TN = 12 and TP = 8, what is the length of PQ?

  1. 8
  2. 10 (correct answer)
  3. 16
  4. 18
Explanation: For a tangent and secant from the same external point, TN² = TP × TQ. So 12² = 8 × TQ, giving 144 = 8 × TQ, thus TQ = 18. Therefore PQ = TQ - TP = 18 - 8 = 10. Choice A incorrectly uses TP. Choice C uses TN as the answer. Choice D incorrectly uses TQ.