ACT Math Practice Question #1163 (Hard (36)) | Test Citadel
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ACT Math Difficulty: Hard (36)

Digital ACT Math Practice Question #1163

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

In triangle \( \triangle PQR \), the length of \( PQ = 7 \), \( QR = 10 \), and the measure of \( \angle PQR = 120^\circ \). What is the exact length of side \( PR \)?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab\cos(C). Remember that cos(120°) = -1/2, making the last term positive.

Elimination Framework (Hint 2)

Desmos Shortcut: Type sqrt(7^2 + 10^2 - 2*7*10*cos(120 deg)) into Desmos. It gives ~14.7986. Check sqrt(219) = 14.7986.

Masterclass Solution & Distractor Trap Analysis

Apply the Law of Cosines: \( PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(120^\circ) \). Since \( \cos(120^\circ) = - rac{1}{2} \), substitute the known values: \[ PR^2 = 7^2 + 10^2 - 2(7)(10)\left(- rac{1}{2} ight) \] \[ PR^2 = 49 + 100 + 70 = 219 \...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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