Pythagorean Theorem

when do you use a + b > c? - can these three sticks physically close to form any triangle?

when do you use a + b = c? - the second you know a 90 degree angle is involved, and you need to calculate for an exact missing length

which tests for existence and which calculates space? - a + b > c for existence; a + b = c for space

if the question asks about: calculating a missing side length - ✅ Pythagorean theorem (a^2 + b^2 = c^2) ❌ SSS/SAS congruence

if the question asks about: proving two triangles are identical copies - ✅ SSS/SAS/ASA/AAS/HL ❌ Pythagorean Theorem

if the question asks about: predicting what type of triangle it is - ✅ Converse Pythagorean Theorem ❌ Angle-Sum Theorem

Direct Pythagorean Theorem - "If it's a right triangle, then a^2 + b^2 = c^2"

Converse Pythagorean Theorem - "Let's look at the side lengths (a^2 + b^2 VS c^2) to see how the triangle behaves"

determining the type of triangle - always square the two smaller sides, add them, and compare them to the square of the longest side (c)

perfect right angle - a^2 + b^2 = c^2; sum is equal to c

acute triangle - a^2 + b^2 > c^2; sum is greater than c, all angles are less than 90 degrees

obtuse triangle - a^2 + b^2 < c^2; sum is less than c, meaning c is wide

famous Pythagorean theorem triple - 5, 12, 13

the longest side of a right triangle - ALWAYS c

4 Types of Geometric Transformations - Translation, Rotation, Reflection, & Dilation

Translation - 👁sliding a shape across a straight line;

Translation - 📏you just add or subtract directly from the x and y coordinates

Rotation - 👁pin the shape on one point on the grid (finding the center of rotation), and spin the entire shape around that;

Rotation - 📏swapping the positions of x and y & changing their signs depending on which way you spin (90, 180, or 270)

Reflection - 👁you pick a straight line on the grid (line of reflection) and treat it like a perfect mirror, you flip it over so i faces the exact opposite direction;

Reflection - 📏flipping over x-axis (x, -y) and flipping over y-axis (-x, y)

Dilation - 👁basically scaling shapes bigger or smaller, keeping its proportions intact;

Dilation - 📏you multiply both coordinates by a scale factor(k)