📌 Step 1: Apply the Triangle Proportionality Theorem
When a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

📌 Step 2: Set up the proportion
AD/DB = AE/EC
6/4 = 9/EC

📌 Step 3: Cross-multiply and solve
6 × EC = 4 × 9
6 × EC = 36
EC = 36/6 = 6

💡 Tip: This theorem is also called the "Side Splitter Theorem." It works because DE ∥ BC creates similar triangles.

📌 Step 1: Identify the right triangle
The ladder, wall, and ground form a right triangle where:
• The wall height = 12 ft (one leg)
• The ground distance = 5 ft (other leg)
• The ladder = hypotenuse (what we need)

📌 Step 2: Apply the Pythagorean Theorem
a² + b² = c²
12² + 5² = c²
144 + 25 = c²
169 = c²

📌 Step 3: Solve for c
c = √169 = 13 feet

💡 Tip: 5-12-13 is a common Pythagorean triple. Memorizing these saves time on the CBE!

📌 Step 1: Recall the distance formula
d = √((x₂ − x₁)² + (y₂ − y₁)²)

📌 Step 2: Plug in the coordinates
A(2, 3) and B(6, 6):
d = √((6 − 2)² + (6 − 3)²)
d = √(4² + 3²)
d = √(16 + 9)

📌 Step 3: Solve
d = √25 = 5

💡 Tip: The distance formula is just the Pythagorean theorem applied to coordinates!

📌 Step 1: Recall the Pythagorean Theorem test
A triangle is a right triangle if and only if a² + b² = c², where c is the longest side.

📌 Step 2: Identify the longest side
The sides are 7, 24, and 25. The longest side is 25.

📌 Step 3: Test the condition
7² + 24² = 49 + 576 = 625
25² = 625

📌 Step 4: Compare
625 = 625 ✓ → Yes, it is a right triangle.

💡 Tip: 7-24-25 is another Pythagorean triple worth memorizing!

📌 Step 1: Recall the Vertical Angles Theorem
When two lines intersect, they form two pairs of vertical angles. Vertical angles are always congruent.

📌 Step 2: Identify the vertical angles
The two angles are across from each other at the intersection point.

📌 Answer: The other vertical angle = 48°

💡 Key Fact: Vertical angles are formed by intersecting lines and are ALWAYS equal — no parallel lines needed! The adjacent angles (linear pair) add up to 180°.

📌 Step 1: Recall the 45-45-90 triangle relationships
In a 45-45-90 triangle:
• Both legs are equal
• Hypotenuse = leg × √2

📌 Step 2: Apply the formula
leg = 8
hypotenuse = 8 × √2 = 8√2 ≈ 11.31

💡 Memory trick: In a 45-45-90 triangle, think "multiply by √2 to get the hypotenuse." For a 30-60-90, the ratios are 1 : √3 : 2.

📌 Step 1: Recall the diagonal formula for a square
For a square with side length s, the diagonal d = s√2.

📌 Step 2: Set up the equation
s√2 = 10√2

📌 Step 3: Solve for s
Divide both sides by √2:
s = 10 cm

💡 Tip: This comes from the 45-45-90 special triangle — every square's diagonal creates two 45-45-90 triangles.

📌 Step 1: Understand dilation
A dilation with scale factor k multiplies every side length by k. Angles stay the same.

📌 Step 2: Apply scale factor of 3
• Side 3 × 3 = 9
• Side 4 × 3 = 12
• Side 5 × 3 = 15

📌 Answer: New sides are 9, 12, 15

💡 Key Facts about dilation:
• Scale factor > 1 → enlargement
• Scale factor < 1 → reduction
• Scale factor = 1 → same size
• Angles NEVER change in a dilation

📌 Step 1: Recall what a contrapositive is
For a conditional "If P, then Q":
• Converse: If Q, then P
• Inverse: If not P, then not Q
• Contrapositive: If not Q, then not P

📌 Step 2: Apply to this statement
Original: "If it rains, then the ground is wet"
Contrapositive: "If the ground is not wet, then it does not rain"

💡 Key Fact: The contrapositive ALWAYS has the same truth value as the original statement. This is a fundamental rule of logic!