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° ✓

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.

📌 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: 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!

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.

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.

📌 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).

📌 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.

📌 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.

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.

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: 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).

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: 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 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: 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).

📌 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 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.