Geometry — Semester A Practice

Question 1 of 10

TEKS 9A-9B Medium 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)

A 169.2 feet

B 47.9 feet

C 203.4 feet

D 101.8 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

TEKS 5A-5D Easy 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?

A 40°

B 75°

C 115°

D 65°

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 3 of 10

TEKS 7A-7B Medium 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?

A 27 feet

B 30 feet

C 36 feet

D 24 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

TEKS 2A-2C Medium Calc Word Diagram

Find the distance between points P and Q shown on the coordinate plane below.

A √13

B √10

C √17

D 5

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 5 of 10

TEKS 7A-7B Medium Calc Word Diagram

In the figure below, DE ∥ BC. If AD = 4, DB = 6, and AE = 5, find EC.

A 7.5

B 6.0

C 8.0

D 10.0

Explanation

📌 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!

Question 6 of 10

TEKS 1A-1G Hard Calc Word

A flagpole casts a shadow 15 feet long. At the same time, a 6-foot person standing nearby casts a shadow 4 feet long. How tall is the flagpole?

A 22.5 feet

B 24.0 feet

C 20.0 feet

D 18.0 feet

Explanation

📌 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.

Question 7 of 10

TEKS 4A-4D Easy Calc Word Diagram

Jake claims: "If a quadrilateral has four right angles, then it must be a square." Which figure below is a counterexample?

A Rectangle

B Trapezoid

C Rhombus

D Square

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 8 of 10

TEKS 1A-1G Medium Calc Word Diagram

A zip-line connects the top of a 40-foot platform to a point on the ground 75 feet away. What is the length of the zip-line cable?

A 80 feet

B 85 feet

C 95 feet

D 75 feet

Explanation

📌 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).

Question 9 of 10

TEKS 8A-8B Hard 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

B 2√15 ≈ 7.75

C √85 ≈ 9.22

D 8.5

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 10 of 10

TEKS 6A-6E Easy Calc Word Diagram

In the triangle below, ∠A = 55° and ∠B = 65°. What is the measure of ∠C?

A 75°

B 70°

C 50°

D 60°

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° ✓

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