Geometry & SAT Vocabulary Tutoring Session Notes

Similar Triangles Refresher

• Definition

  • Two triangles are similar when all corresponding angles are equal and all corresponding sides are scaled by the same constant factor (the similarity ratio).
  • Informal student description: “Triangles that get bigger/smaller through scaling; the angles stay, the lengths stretch/shrink together.”

• Key properties

  • Corresponding angle pairs ≅ (∠A ≅ ∠A′, etc.)
  • Corresponding side ratios constant: \frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}=k (where k is the scale factor)
  • If one side is doubled, every side is doubled; if one side is multiplied by \frac{3}{2} every side is.

• Example worked in session

  • Small triangle sides: 8,\;10,\;15 (labelled in discussion)
  • Partner larger triangle side known: 12 corresponds to 8. Scale factor k=\frac{12}{8}=\frac{3}{2}.
  • Unknown corresponding long side found: 15 \times \frac{3}{2}=22.5.
  • Alternate phrasing used by student: “12 is 50 % more than 8, so multiply 15 by 1.5 ⇒ 22.5.”

• Ratio statement emphasized for SAT work

  • Any ratio of corresponding sides in similar triangles is equal, e.g. \frac{\text{short}1}{\text{short}2}=\frac{\text{hyp}1}{\text{hyp}2}.

Angle Relationships in Parallel‐Line Diagrams

• Vocabulary recalled

  • Corresponding angles
  • Alternate interior angles
  • Vertical angles

• Straight-line and triangle angle sums

  • Straight line: adjacent angles add to 180^{\circ}.
  • Triangle: interior angles sum to 180^{\circ}.

• Quick example discussed

  • Given parallel lines produced three equal angles labelled 58^{\circ},\;74^{\circ} and asked to solve e & z.
  • Derived equation: 74+58+e=180 \Rightarrow e=48^{\circ} (value verbally hinted, not solved on screen).
  • Method 2: use triangle sum with one angle 58^{\circ}, second 74^{\circ}, third z=48^{\circ}.

SAT Vocabulary Practice & Context Clues

• Passage topic: traumatic memories and stress hormones (epinephrine, cortisol) leading to “_ memory and disrupted chronology.”
• Unknown words for student: fractious, mendacious, evanescent. Known: inane.
• Reasoning
  • Look for root clues: “fract-” relates to “fracture/fraction → broken.”
  • Chosen answer: fractious (incorrect by dictionary, but supported by root logic).
• Correct meanings supplied
  • fractious – irritable, quarrelsome (esp. children)
  • mendacious – lying, untruthful
  • evanescent – fleeting, vanishing
  • inane – silly, empty of meaning

Second vocabulary item

• Sentence about Meso-American earth monster: “a sacrosanct _ being.” Choices included benign & abominable.
• Definitions clarified
  • benign – harmless, non-malignant (cf. “benign tumor”)
  • abominable – detestable, disgusting (root “abomination”)
• Logical fit: sacred + living → benign (emotion-neutral). Student chose abominable; tutor explained why benign is more precise.
• New word defined
  • sacrosanct – sacred, inviolable (root “sacred/sanctified”).

Triangle Classifications (by Sides & Angles)

• By sides

  • Equilateral: all three sides equal (also automatically isosceles).
  • Isosceles: at least two sides equal; base angles opposite equal sides are equal.
  • Scalene: all three sides unequal.

• By angles

  • Acute: all three interior angles <90^{\circ}.
  • Right: exactly one 90^{\circ} angle.
  • Obtuse: exactly one angle >90^{\circ}.
  • Terminology inside right triangle
  • Legs: the two sides adjacent to the right angle.
  • Hypotenuse: side opposite the right angle.
  • Pythagorean Theorem referenced: a^{2}+b^{2}=c^{2}.

Special Right Triangles & Why They Matter

• 45^{\circ}!–!45^{\circ}!–!90^{\circ} (Isosceles Right)

  • Side ratio: x : x : x\sqrt{2}.
  • Visual origin: diagonal of a square of side x.

• 30^{\circ}!–!60^{\circ}!–!90^{\circ}

  • Side ratio: x : x\sqrt{3} : 2x (across 30^{\circ} is the shortest x, across 90^{\circ} is 2x).
  • Visual origin: height dropped in an equilateral triangle splits it into two 30\text{–}60\text{–}90 triangles.

• Application example

  • Equilateral triangle with side 6.
  • Height =3\sqrt{3} (from 30\text{–}60\text{–}90 ratio: x=3 so altitude x\sqrt{3}=3\sqrt{3}).
  • Area formula A=\frac12 bh = \frac12 (6)(3\sqrt{3}) = 9\sqrt{3} \approx 15.6.

Altitude in Right Triangles → Three Similar Triangles

• Dropping an altitude from right angle to hypotenuse creates two smaller right triangles; both are similar to each other and to the original.

• Angle proof sketched

  • Original right angle =90^{\circ}.
  • Label non-right angles x and 90^{\circ}-x.
  • All three triangles share angles x,\;90^{\circ}-x,\;90^{\circ} → similarity established.

• Typical SAT setup from session

  • Small triangle: short side 3, hypotenuse 5.
  • Large triangle: hypotenuse (3+w).
  • Corresponding-side proportion: \frac{3}{\text{short}_{\text{big}}}=\frac{5}{3+w}.
  • Cross-multiply: 9+3w=25 \Rightarrow w=\frac{16}{3}\;(\approx5.33).
  • Tutor highlighted this as a very common SAT geometry question.

Practice Problem Snippets Mentioned

• Parallel-line puzzle: multiple symmetries; angles 76^{\circ} and 21^{\circ} placed; recognizing mirror/rotational symmetry to equate angles.
• Misc. suggestion: copy altitude-similar-triangle derivation into study notes for future reference.

Test-Taking & Study Tips Embedded

• When unknown vocabulary appears, hunt for roots & context (e.g., “fract-” suggests “broken”).
• For adjective lists separated by commas, order can be swapped or joined by “and” → adjectives coordinate, not cumulative.
• Remember special-right-triangle ratios cold; they appear in square diagonals, equilateral-triangle heights, and SAT coordinate geometry.
• Keep a geometry formula sheet handy (altitude similarity, area shortcuts, Pythagorean triples).

Miscellaneous Session Moments

• Student did some homework the previous evening and in the morning.
• Parallel-line/angle reasoning previously covered was reviewed quickly.
• Student earned a 5 on AP World History; tutor congratulated.
• Upcoming football practice reminded student of time constraints.