Find the distance between points P and Q shown on the coordinate plane below.
A5
B√10
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 CBE.
Question 2 of 10
TEKS 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?
A24 feet
B30 feet
C36 feet
D27 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 3 of 10
TEKS 6A-6EEasy Calc Word Diagram
In the triangle below, ∠A = 55° and ∠B = 65°. What is the measure of ∠C?
A75°
B70°
C50°
D60°
Explanation
📌 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° ✓
Question 4 of 10
TEKS 9A-9BMedium Calc Word Diagram
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)
A169.2 feet
B47.9 feet
C101.8 feet
D203.4 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 5 of 10
TEKS 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?
A65°
B115°
C40°
D75°
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
TEKS 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?
ARhombus
BRectangle
CSquare
DTrapezoid
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 7 of 10
TEKS 8A-8BHard Calc Word Diagram
In right triangle ABC, an altitude CD is drawn from the right angle C to hypotenuse AB. If AD = 5 and DB = 12, what is the length of CD?
A√17 ≈ 4.12
B2√15 ≈ 7.75
C8.5
D√85 ≈ 9.22
Explanation
The altitude to the hypotenuse is the geometric mean of the two segments: CD = √(AD × DB) = √(5 × 12) = √60 = 2√15 ≈ 7.75.
Question 8 of 10
TEKS 7A-7BMedium Calc Word Diagram
In the figure below, DE ∥ BC. If AD = 4, DB = 6, and AE = 5, find EC.
A8.0
B7.5
C6.0
D10.0
Explanation
📌 Step 1: Apply the Triangle Proportionality Theorem Since DE ∥ BC: AD/DB = AE/EC