2025년 백마고 1 2학기 중간고사 해설

Main ideas / lessons

• The speaker walks through solutions to multiple problems from “2025 Baengmago (백마고) 1st Year, 2nd Semester Midterm Exam”, using standard coordinate-geometry, circle properties, and set theory techniques.
• Recurring themes:
  • Find circle centers and radii from given diameter endpoints and use distance formulas.
  • Use coordinate transformations (translations/reflections) to apply rules to points/graphs.
  • Use line relationships (perpendicular/parallel slopes; angle bisectors; distance from a point to a line).
  • Optimize geometry problems by recognizing extremal configurations (e.g., maximal area occurs when a line passes through key circle geometry like radius/diameter).
  • Use set complements and inclusion-exclusion style counting with modular arithmetic periodicity.
  • Use symmetry/reflection to convert path/minimum-length problems into straight-line distance.
  • Convert tangent/line conditions to circle equations (standard vs. tangent form).

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Detailed methodology / instructions (as presented)

Circle geometry (center/radius, (r^2) expressions)

Diameter endpoints → center and radius

• When two points are the endpoints of a diameter:

  • Center = midpoint of the two points:

    • [ a = \frac{x_1 + x_2}{2}, \quad b = \frac{y_1 + y_2}{2} ]

  • Radius = half the distance between diameter endpoints.

  • If the problem asks for (r^2), compute:

    • [ r^2 = \frac{(x_1-x_2)^2}{4} + \frac{(y_1-y_2)^2}{4} ]

    • (The speaker describes a version of this by effectively subtracting squared coordinate differences.)

Circle tangent to the x-axis and y-axis

• For two circles tangent to the x-axis and y-axis:

  • If a circle is tangent to both axes in the first quadrant, its center has the form ((r, r)).

  • Use the condition that the circle passes through a point ((x, y)):

    • [ (x-r)^2 + (y-r)^2 = r^2 ]

  • Solve for (r) (or the coordinate form being used), then compute the distance between centers using the point-distance formula.

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Line relationships (perpendicular vs. parallel)

• For lines in slope-intercept form:
  • Perpendicular lines: slope product (= -1).
  • Parallel lines: slopes are equal.
• The speaker emphasizes careful checking when intercepts differ.

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Translating coordinate graphs

• To apply a translation “move by ((\Delta x, \Delta y))”:
  • Take a point ((x,y)) in the original expression and replace it with the shifted coordinates.
  • Effectively substitute the translated point into the given curve equation.
  • Determine parameters (e.g., solve for (a) from the resulting substitution).

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Angle bisector / dividing an angle (slope approach)

• To find a line that bisects the acute angle between a given line (L_1) and the x-axis:
  • Identify the line(s) forming the right-angle structure using geometry (the speaker uses a right-triangle observation from intercepts).
  • Use the property that angle bisectors create equal angle regions.
  • In slope form, the bisector line can be derived via a structured method or distance-to-line reasoning.
  • The speaker ultimately uses a ratio-based / triangle decomposition approach to obtain the bisector’s slope.

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Reflections to solve minimum path (“broken line”) problems

• Standard method used:
  1. Reflect the figure across the relevant axis (or line) to “unfold” the path.
  2. Convert the bent/reflected path into a straight line between an original point and a reflected point.
  3. If there are multiple reflections, apply them sequentially.
  4. Compute the straight-line distance using the distance formula or Pythagorean theorem.

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Repeated point transformations (cycles)

• For iterated reflections depending on quadrants:
  • Determine the rule per quadrant (e.g., reflect across y-axis, swap x/y, reflect across origin, etc.).
  • Track how many steps are needed for a full cycle (the speaker states it repeats in groups of 8).
  • Reduce work using modular arithmetic:
    • Compute (33 \bmod 8) to perform only the remaining steps after removing full cycles.

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Circle tangent problems

Tangent line given at a point on the circle

• For a tangent line given at a point on the circle:

  • If the tangent line is written in a transformed/shifted form, first convert the circle equation into a compatible standard form (express center/radius appropriately).

  • Use the condition:

    • Distance from center to tangent line equals the radius:
      • [ \text{dist}((h,k), \text{line}) = r ]

  • Solve the resulting algebra for the unknown parameter(s).

Circle tangent to a line at a point

• For “circle tangent to a line at a point”:
  • Use the tangent line equation at that point and apply the distance-to-line condition.

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Set theory / counting with modular conditions (complements)

• For statements like:
  • “remainder mod 4 is not 2” OR “remainder mod 6 is not 4”
• Speaker’s approach:
  • Use complement events:
    • Count total (\le 200), then subtract the count where the undesired complementary condition fails.
  • Let:
    • (A): numbers with remainder 2 when divided by 4
    • (B): numbers with remainder 4 when divided by 6
  • The desired count corresponds to a form like:
    • (\text{Total} - |A \cap B|), with additional inclusion-exclusion handling for overlaps.
  • Use arithmetic progression intersection via LCM periodicity.

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Speaker / sources featured

• Single speaker/teacher (no other named speakers or external sources mentioned in the subtitles).