From the top of a lighthouse 90 feet tall, the angle of depression to a boat is 28°. How far is the boat from the base of the lighthouse? (tan 28° ≈ 0.532)
A203.4 feet
B169.2 feet
C47.9 feet
D101.8 feet
Explanation
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.
Question 2 of 10
Massachusetts 2A-2CMedium Calc Word Diagram
Find the distance between points P and Q shown on the coordinate plane below.
A√10
B5
C√13
D√17
Explanation
📌 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 Regents.
Question 3 of 10
Massachusetts 7A-7BMedium Calc Word Diagram
A tree casts a shadow 18 feet long. At the same time, a 5-foot-tall fence post casts a shadow 3 feet long. How tall is the tree?
A36 feet
B27 feet
C24 feet
D30 feet
Explanation
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.
Question 4 of 10
Massachusetts 4A-4DEasy Calc Word Diagram
Jake claims: "If a quadrilateral has four right angles, then it must be a square." Which figure below is a counterexample?
ATrapezoid
BRectangle
CSquare
DRhombus
Explanation
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.
Question 5 of 10
Massachusetts 5A-5DEasy Calc Word Diagram
The exterior angle of a triangle is 140°. One of the non-adjacent interior angles is 65°. What is the other non-adjacent interior angle?
A115°
B65°
C75°
D40°
Explanation
📌 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.
Question 6 of 10
Massachusetts 1A-1GEasy Calc Word Diagram
A wheelchair ramp must have a slope ratio of 1:12 (rise:run). If the entrance is 2.5 feet above the ground, how long must the ramp be along the ground?
A36 feet
B24 feet
C30 feet
D28 feet
Explanation
📌 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 Regents.
Question 7 of 10
Massachusetts 7A-7BMedium Calc Word Diagram
In the figure below, DE ∥ BC. If AD = 4, DB = 6, and AE = 5, find EC.
A7.5
B6.0
C10.0
D8.0
Explanation
📌 Step 1: Apply the Triangle Proportionality Theorem Since DE ∥ BC: AD/DB = AE/EC