ACT Math Practice Question #853 (Hard (36)) | Test Citadel
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ACT Math Difficulty: Hard (36)

Digital ACT Math Practice Question #853

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

In triangle \( ABC \), side \( AB \) has length 7 meters, side \( BC \) has length 8 meters, and \( \cos(\angle B) = \frac{1}{4} \). What is the exact length, in meters, of side \( AC \)?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

When given two sides and the included angle, apply the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \).

Elimination Framework (Hint 2)

TI-84 / Calculator Shortcut: Compute \( 7^2 + 8^2 - 2 \times 7 \times 8 \times 0.25 = 85 \). The answer is simply \( \sqrt{85} \).

Masterclass Solution & Distractor Trap Analysis

Use the Law of Cosines: \( b^2 = a^2 + c^2 - 2ac\cos(C) \). Here, \( a = BC = 8 \), \( c = AB = 7 \), and \( \cos(C) = \frac{1}{4} \): \[ AC^2 = 8^2 + 7^2 - 2(8)(7)\left(\frac{1}{4}\right) \] \[ AC^2 = 64 + 49 - 112\left(\frac{1}{4}\right) ...

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

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