NECO 2001 Mathematics · Question 15

In an acute angle-triangle XYZ, XZ = 9cm, YZ = 11cm and ∠XZY = 35° Calculate ∠YXZ, correct to the nearest degree.

Options

A.57°
B.45°
C.33°
D.16°
E.89°

Answer: E. 89°

Explanation written with AI and checked against the answer key

Analyze the question: We know two sides (XZ = 9, YZ = 11) and the included angle Z = 35°. We need the angle at X, which is opposite side YZ. Concept: Use the cosine rule to find the third side XY, then the sine rule to find the angle. Solution: Step 1: Find XY (opposite Z). XY² = 9² + 11² − 2(9)(11)cos 35° XY² = 81 + 121 − 198(0.8192) XY² = 202 − 162.20 = 39.80 XY ≈ 6.31 cm Step 2: Use the sine rule to find ∠YXZ (opposite YZ = 11). sin X / 11 = sin 35° / 6.31 sin X = 11(0.5736) / 6.31 = 6.3096 / 6.31 ≈ 0.99997 So X ≈ 90° (or 89.5° to more precision), which makes sense: the triangle is nearly right-angled at X. Let's check precisely: cos X = (XZ² + XY² − YZ²)/(2·XZ·XY) = (81 + 39.80 − 121)/(2·9·6.309) = (−0.20)/113.6 ≈ −0.0018, so X ≈ 90.1°. The exact value is around 90°, which is not acute, yet the question says the triangle is acute. Rounded to the nearest degree, the computed answer is 90°, and the closest option is E (89°). The wording "acute-angled" in the question is slightly inconsistent with the data (the angle is essentially 90°), but E is the only option that fits the calculation. Also check the other angle: Y = 180 − 35 − 90 ≈ 55°, which is near option A (57°), but that is angle Y, not angle YXZ, so it is a trap. Answer: E (89°) Tip: Always identify which angle is opposite which side before using the sine rule. Here, ∠YXZ is opposite YZ. Also, if the sine of an angle comes out near 1, the angle is close to 90°.

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