Geometry — Semester A Practice

Question 1 of 10

TEKS 1A-1GHard 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?

A22.5 feet

B20.0 feet

C24.0 feet

D18.0 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 2 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°

C60°

D50°

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 3 of 10

TEKS 1A-1GMedium 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?

A80 feet

B85 feet

C95 feet

D75 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 4 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?

A2√15 ≈ 7.75

B8.5

C√85 ≈ 9.22

D√17 ≈ 4.12

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 5 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

BSquare

CRectangle

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 6 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?

A75°

B40°

C65°

D115°

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 7 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)

A47.9 feet

B169.2 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 8 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?

A30 feet

B36 feet

C27 feet

D24 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 9 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

C10.0

D6.0

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 10 of 10

TEKS 2A-2CMedium Calc Word Diagram

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

A√13

C√17

D√10

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.

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