In the xy-plane, line j passes through the origin and the point \(\mathrm{(c, 1)}\), where c is a positive constant....
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the \(\mathrm{xy}\)-plane, line \(\mathrm{j}\) passes through the origin and the point \(\mathrm{(c, 1)}\), where \(\mathrm{c}\) is a positive constant. The acute angle between line \(\mathrm{j}\) and the positive \(\mathrm{x}\)-axis is \(\mathrm{p°}\). Line \(\mathrm{k}\) passes through the origin and the point \(\mathrm{(1, c)}\). The acute angle between line \(\mathrm{k}\) and the positive \(\mathrm{x}\)-axis is \(\mathrm{q°}\). Which of the following correctly expresses \(\mathrm{q}\) in terms of \(\mathrm{p}\)?
\(\mathrm{p}°\)
\((90-\mathrm{p})°\)
\((90+\mathrm{p})°\)
\((180-\mathrm{p})°\)
1. TRANSLATE the geometric information into slopes
- Given information:
- Line j passes through \((0,0)\) and \((c,1)\), makes acute angle \(p°\) with positive x-axis
- Line k passes through \((0,0)\) and \((1,c)\), makes acute angle \(q°\) with positive x-axis
- c is positive
- What this tells us:
- Slope of line j = \(\frac{1}{c}\)
- Slope of line k = \(\frac{c}{1} = c\)
2. INFER the angle-slope relationships
- For lines through the origin making acute angles with the positive x-axis, the tangent of the angle equals the slope
- Therefore: \(\tan(p°) = \frac{1}{c}\) and \(\tan(q°) = c\)
3. INFER the key relationship between the angles
- Notice that \(\tan(p°) = \frac{1}{c}\) and \(\tan(q°) = c\)
- This means: \(\tan(p°) = \frac{1}{\tan(q°)} = \cot(q°)\)
4. INFER using trigonometric identities
- Apply the identity: \(\cot(x) = \tan(90° - x)\)
- Therefore: \(\tan(p°) = \tan(90° - q°)\)
- Since both angles are acute: \(p = 90 - q\)
5. SIMPLIFY to find the final relationship
- From \(p = 90 - q\), we get: \(q = 90 - p\)
Answer: B. \((90-p)°\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students recognize that the slopes are \(\frac{1}{c}\) and \(c\) but fail to see that these are reciprocals or don't connect this to the cotangent relationship.
Without this key insight, they might guess that the angles are equal since both lines pass through the origin, leading them to select Choice A (\(p°\)).
Second Most Common Error:
Missing conceptual knowledge: Students don't remember or incorrectly apply the cotangent identity \(\cot(x) = \tan(90° - x)\).
They might think the relationship involves supplementary angles (adding to 180°) rather than complementary angles (adding to 90°), leading them to select Choice D (\((180-p)°\)).
The Bottom Line:
This problem tests whether students can recognize reciprocal relationships in slopes and connect them to complementary angle relationships through trigonometric identities. The key insight is seeing that when two lines through the origin have reciprocal slopes, their angles with the x-axis are complementary.
\(\mathrm{p}°\)
\((90-\mathrm{p})°\)
\((90+\mathrm{p})°\)
\((180-\mathrm{p})°\)