Geometry Notes: Square Verification and Law of Cosines

Square checks and geometric verification

  • Hint from the slide: when checking if a quadrilateral is a square, the first instinct is to check the longest side.
  • Practical approach mentioned:
    • Measure from one corner to the opposite corner (diagonal) as a quick check.
    • You could also measure from another corner and verify the following:
    • The two outside (exterior) angles are correct.
    • The perimeter is consistent with a square.
  • Critical question raised: even if these checks pass (diagonal length, exterior angles, and perimeter), could you still have angles that are not a square?
  • Visual cues described:
    • In the shadow, there is a specific top point and a blade that is straight up.
    • Commentary note: we don’t know certain details, which leads to following steps in reasoning.
  • Core takeaway: to certify a square, you should establish both equal side lengths and right angles; merely matching one or two conditions is not sufficient.
  • Quick attribute recap for a square (foundation, not explicitly enumerated in the slide but relevant to the topic):
    • All four sides equal: if the side length is s, then the perimeter P satisfies
      P=4s.P = 4s.
    • All interior angles are right angles: each measures 90∘90^{\circ}.
    • Diagonals are equal and perpendicular.

Triangle angle calculation when all three sides are known

  • Scenario described: you have a triangle with all three sides known and you’re asked to find angle $A$ (denoted as the angle opposite side $a$).
  • Three distinct approaches are possible when all sides are known (as mentioned in the transcript):
    • Approach 1: Law of Cosines directly to solve for the angle.
    • Approach 2: Convert to a cosine expression and then take the inverse cosine to obtain the angle.
    • Approach 3: Use the same Law of Cosines in a symmetric fashion to obtain any of the three angles (A, B, or C) by substituting the appropriate opposite side.
  • Law of Cosines (for angle $A$ opposite side $a$):
    • Formula:
      cos⁡A=b2+c2−a22bc\boxed{\cos A = \frac{b^2 + c^2 - a^2}{2bc}}
    • Consequently,
      A=cos⁡−1(b2+c2−a22bc).\boxed{A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)}.
  • TI calculator usage notes (as described):
    • Before computing, set the calculator to the correct angle mode (degrees, not radians).
    • On TI calculators, this is typically done by:
    • Accessing the Mode/Angle settings and selecting Degrees (DEG) instead of Radians (RAD).
    • The transcript notes this is found in a third-menu-down path, under the option for Radiant or Degree; ensure Degree is selected.
    • Computation steps example (in general form):
    • Compute the inside of the cosine:
      b2+c2−a22bc.\frac{b^2 + c^2 - a^2}{2bc}.
    • Then compute the angle using the inverse cosine:
      θ=cos⁡−1(b2+c2−a22bc).\theta = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right).
    • On the TI, you would input something like:
    • acos((b^2 + c^2 - a^2)/(2bc)) and ensure the calculator is in Degrees mode, so the result is in degrees.
  • Important conceptual point: with all three sides known, you can compute any angle using the Law of Cosines by plugging in the appropriate $a$, $b$, and $c$.
  • Explicit caveat mentioned in the transcript: when using a calculator, attention must be paid to whether you’re in radians or degrees; this affects the numeric result.

Illustrated example setup (conceptual, based on transcript proportions)

  • If you have a triangle with sides labeled $a$, $b$, and $c$, and you’re solving for angle $A$ opposite side $a$:
    • Use:
      cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}
    • Then:
      A=cos⁡−1(b2+c2−a22bc).A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right).
  • Practical notes for exam preparation:
    • Always verify the angle unit mode on your calculator before computing inverse trigonometric functions.
    • When writing notes, include both the formula and the step-by-step calculation pattern so you can reproduce the method quickly on an exam.
    • If you know all three sides, you have three equivalent ways to compute the three possible angles using the same Law of Cosines framework by cycling which side is treated as opposite the angle you want to find.

Summary of key concepts and practical implications

  • Quick square-check protocol emphasizes starting with the longest side as a heuristic; however, a complete verification requires equal sides and right angles.
  • Exterior angle and perimeter checks can help identify potential non-square configurations, but passing these checks does not guarantee a square.
  • In triangle problems with all sides known, the Law of Cosines is the primary tool to determine any angle, with the inverse cosine giving the angle value in the correct units.
  • Calculator proficiency matters: ensure degree mode is active when calculating angles in degrees, and understand the path to access radians vs degrees on your device.