Geometry — Semester 1 Practice — Michigan Test Out™

Question 1 of 10

Michigan Standards 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 2 of 10

Michigan Standards 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?

A28 feet

B36 feet

C30 feet

D24 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 Test Out.

Question 3 of 10

Michigan Standards 7A-7BMedium Calc Word Diagram

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

A6.0

B8.0

C10.0

D7.5

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 4 of 10

Michigan Standards 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

B√17 ≈ 4.12

C√85 ≈ 9.22

D8.5

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 5 of 10

Michigan Standards 2A-2CMedium Calc Word Diagram

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

A√17

B√10

C√13

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 Test Out.

Question 6 of 10

Michigan Standards 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

BRhombus

CSquare

DRectangle

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

Michigan Standards 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°

B75°

C40°

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 8 of 10

Michigan Standards 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

B203.4 feet

C101.8 feet

D169.2 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 9 of 10

Michigan Standards 6A-6EEasy Calc Word Diagram

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

A75°

B60°

C50°

D70°

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 10 of 10

Michigan Standards 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

B30 feet

C24 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.

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