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 altitude to the hypotenuse is the geometric mean of the two segments:
CD = √(AD × DB) = √(5 × 12) = √60 = 2√15 ≈ 7.75.

📌 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: 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 EOC.

Rectangle area = 20 × 10 = 200 m².
Two semicircles = one full circle with r = 5: π × 5² = 3.14 × 25 = 78.5 m².
Total = 200 + 78.5 = 278.5 m².

📌 Step 1: Identify the shape
A pizza box is a rectangular prism (cuboid).

📌 Step 2: Apply the volume formula
V = length × width × height
V = 14 × 14 × 2

📌 Step 3: Calculate
= 392 in³

💡 Quick check: Volume is always in cubic units. If your answer is in square units, something went wrong!

A quadrilateral with exactly one pair of parallel sides is a trapezoid.
AB ∥ DC but AB ≠ DC (16 ≠ 22), confirming it is a trapezoid, not a parallelogram.

📌 Step 1: Recognize similar triangles
The sun creates the same angle for both the flagpole and the person, making two similar triangles.

📌 Step 2: Set up the proportion
flagpole height / flagpole shadow = person height / person shadow
h / 15 = 6 / 4

📌 Step 3: Cross-multiply and solve
h × 4 = 15 × 6
4h = 90
h = 22.5 feet

💡 Tip: Shadow problems always use similar triangles because the sun's rays are parallel.

📌 Step 1: Recall the volume formula for a cylinder
V = πr²h

📌 Step 2: Plug in the values
r = 3 ft, h = 8 ft, π ≈ 3.14
V = 3.14 × 3² × 8 = 3.14 × 9 × 8

📌 Step 3: Calculate
3.14 × 9 = 28.26
28.26 × 8 = 226.08 ft³

💡 Tip: Always check your units — volume is measured in cubic units (ft³, cm³, m³).

📌 Step 1: Identify the trig ratio
We know the hypotenuse (200 ft) and want the opposite side (height).
Use sine: sin = opposite / hypotenuse

📌 Step 2: Set up and solve
sin(55°) = h / 200
0.819 = h / 200
h = 200 × 0.819 = 163.8

📌 Answer: ≈ 164 feet

💡 Tip: Angle of elevation from ground = angle between the string and the horizontal, NOT the vertical.

📌 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: Check each figure

A (Scalene triangle): No lines of symmetry, no rotational symmetry ✗
B (Regular hexagon): 6 lines of symmetry + rotational symmetry at 60° ✓
C (Parallelogram): No lines of symmetry, rotational symmetry at 180° only → partial ✗
D (Arrow): 1 line of symmetry (vertical) but no rotational symmetry ✗

📌 Answer: B (Regular hexagon)

💡 Tip: All regular polygons have BOTH reflectional AND rotational symmetry. The number of symmetry lines = number of sides.

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 cone volume formula
V = (1/3)πr²h

📌 Step 2: Plug in values
r = 6 cm, h = 15 cm
V = (1/3)(3.14)(36)(15)

📌 Step 3: Calculate step by step
3.14 × 36 = 113.04
113.04 × 15 = 1695.6
1695.6 / 3 = 565.2 cm³

💡 Common mistake: Don't forget to divide by 3! A cone is 1/3 the volume of a cylinder with the same base and height.

📌 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: 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: Break into simple shapes
Rectangle: 10 m × 6 m
Semicircle: radius = 6/2 = 3 m (attached to the 6 m side)

📌 Step 2: Calculate each area
Rectangle = 10 × 6 = 60 m²
Semicircle = ½πr² = ½ × 3.14 × 3² = ½ × 28.26 = 14.13 m²

📌 Step 3: Add them
Total = 60 + 14.13 = 74.13 ≈ 74.1 m²

💡 Strategy for composite figures: Always break them into shapes you know (rectangles, triangles, circles), calculate each, then add (or subtract for holes).

The tangent is perpendicular to the radius at the point of tangency. Using the Pythagorean theorem: PT = √(OP² − OT²) = √(13² − 5²) = √(169 − 25) = √144 = 12.

📌 Step 1: Understand rigid motions (isometries)
Transformations that preserve BOTH size and shape:
✅ Translation (slide)
✅ Reflection (flip)
✅ Rotation (turn)

📌 Step 2: Non-rigid transformations
❌ Dilation — changes size
❌ Stretches/compressions — distort shape

📌 Answer: Translation preserves both size and shape.

💡 Key term: Rigid motions are also called "isometries" (iso = same, metry = measure).