📌 Step 1: Apply the Triangle Proportionality Theorem
Since DE ∥ BC: AD/DB = AE/EC

📌 Step 2: Set up the proportion
4/6 = 5/EC

📌 Step 3: Cross-multiply and solve
4 × EC = 6 × 5 = 30
EC = 30/4 = 7.5

💡 Verification: AD/DB = 4/6 = 2/3. AE/EC = 5/7.5 = 2/3. ✓ The ratios match!

📌 Step 1: Identify the right triangle
The platform height (40 ft), ground distance (75 ft), and cable form a right triangle.

📌 Step 2: Apply the Pythagorean Theorem
cable² = 40² + 75²
cable² = 1600 + 5625
cable² = 7225

📌 Step 3: Solve
cable = √7225 = 85 ft

💡 Tip: This is a multiple of the 8-15-17 Pythagorean triple (×5 = 40-75-85).

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

📌 Step 2: Plug in P(1, 2) and Q(−1, −1)
d = √((1 − (−1))² + (2 − (−1))²)
= √(2² + 3²)
= √(4 + 9)

📌 Answer: d = √13 ≈ 3.61

💡 Tip: Leave your answer in √ form when exact values are expected on the Test Out.

A rectangle has four right angles but is NOT necessarily a square (it can have unequal side lengths).
The rectangle with sides 90×60 is a counterexample to Jake's claim.

The tree and fence post form similar triangles with their shadows (same sun angle).
tree height / tree shadow = fence height / fence shadow
h / 18 = 5 / 3
h = 18 × 5/3 = 30 feet.

The angle of depression equals the angle of elevation from the boat.
tan(28°) = opposite/adjacent = 90/d
d = 90/tan(28°) = 90/0.532 ≈ 169.2 feet.

📌 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 Exterior Angle Theorem
The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

📌 Step 2: Set up the equation
exterior angle = angle A + angle C
140° = 65° + angle C

📌 Step 3: Solve
angle C = 140° − 65° = 75°

💡 Tip: The Exterior Angle Theorem is a shortcut! You don't need to find the interior angle at B first. The exterior angle always equals the sum of the two "remote" interior angles.

📌 Step 1: Understand the slope ratio
1:12 means for every 1 foot of rise, you need 12 feet of run.

📌 Step 2: Set up the proportion
rise/run = 1/12
2.5/run = 1/12

📌 Step 3: Solve
run = 2.5 × 12 = 30 feet

💡 Real-world context: The 1:12 slope ratio is required by the ADA (Americans with Disabilities Act) for wheelchair accessibility. This is a common real-world application tested on the Test Out.