📌 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 the Alternate Interior Angles Theorem
When two parallel lines are cut by a transversal, alternate interior angles are congruent (equal).

📌 Step 2: Identify the angle pair
The two angles are on opposite sides of the transversal and between the parallel lines → they are alternate interior angles.

📌 Step 3: Apply the theorem
Since the lines are parallel, the other angle = 65°

💡 Key Fact: There are 3 angle pairs to know for parallel lines:
• Alternate interior angles → equal
• Corresponding angles → equal
• Co-interior (same-side) angles → supplementary (add to 180°)

📌 Step 1: Recall the Triangle Angle Sum Theorem
All angles in a triangle add up to 180°.

📌 Step 2: Set up the equation
∠A + ∠B + ∠C = 180°
55° + 65° + ∠C = 180°

📌 Step 3: Solve
∠C = 180° − 55° − 65° = 60°

💡 Quick check: 55 + 65 + 60 = 180° ✓

📌 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: 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: 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: Recall the perpendicular slope rule
If two lines are perpendicular, their slopes are negative reciprocals of each other.

📌 Step 2: Find the negative reciprocal
Original slope = 3/4
• Flip it: 4/3
• Negate it: −4/3

📌 Answer: The perpendicular slope is −4/3

💡 Tip: Multiply perpendicular slopes together and you always get −1: (3/4)(−4/3) = −1 ✓

📌 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!