Geometry — Semester 2 Practice — Michigan Test Out™

Question 1 of 10

Michigan Standards 11A-11DMedium Calc Word Diagram

Find the volume of the cone shown below. Round to the nearest tenth. (Use π ≈ 3.14)

A452.2 cm³

B1695.6 cm³

C565.2 cm³

D339.1 cm³

Explanation

📌 Step 1: Recall the cone volume formula
V = (1/3)πr²h

📌 Step 2: Plug in values
r = 6 cm, h = 15 cm
V = (1/3)(3.14)(36)(15)

📌 Step 3: Calculate step by step
3.14 × 36 = 113.04
113.04 × 15 = 1695.6
1695.6 / 3 = 565.2 cm³

💡 Common mistake: Don't forget to divide by 3! A cone is 1/3 the volume of a cylinder with the same base and height.

Question 2 of 10

Michigan Standards 1A-1GMedium Calc Word Diagram

Quadrilateral ABCD has the properties shown below. Which type of quadrilateral is ABCD?

ARectangle

BTrapezoid

CParallelogram

DRhombus

Explanation

A quadrilateral with exactly one pair of parallel sides is a trapezoid.
AB ∥ DC but AB ≠ DC (16 ≠ 22), confirming it is a trapezoid, not a parallelogram.

Question 3 of 10

Michigan Standards 10A-10BMedium Calc Word Diagram

A cylinder has radius 4 and height 10. If the radius is tripled but the height stays the same, by what factor does the volume increase?

B27

Explanation

📌 Step 1: Recall how dimensional changes affect volume
V = πr²h
When r is multiplied by k (and h stays the same):
New V = π(kr)²h = k²πr²h = k² × original V

📌 Step 2: Apply k = 3
Factor = 3² = 9

📌 Verification:
Original: π(4²)(10) = 160π
New: π(12²)(10) = 1440π
1440π / 160π = 9 ✓

💡 Key rule: Changing ONE linear dimension by factor k changes:
• Length → k
• Area → k²
• Volume → k² (if only radius) or k³ (if all dimensions)

Question 4 of 10

Michigan Standards 1A-1GMedium Calc Word Diagram

A kite is flying at the end of a 200-foot string. The string makes a 55° angle with the ground. How high above the ground is the kite? Round to the nearest foot. (sin 55° ≈ 0.819)

A186 feet

B141 feet

C164 feet

D115 feet

Explanation

📌 Step 1: Identify the trig ratio
We know the hypotenuse (200 ft) and want the opposite side (height).
Use sine: sin = opposite / hypotenuse

📌 Step 2: Set up and solve
sin(55°) = h / 200
0.819 = h / 200
h = 200 × 0.819 = 163.8

📌 Answer: ≈ 164 feet

💡 Tip: Angle of elevation from ground = angle between the string and the horizontal, NOT the vertical.

Question 5 of 10

Michigan Standards 10A-10BMedium Calc Word Diagram

A plane cuts through a right square pyramid parallel to the base, as shown. What is the shape of the cross section?

ATriangle

BPentagon

CSquare

DCircle

Explanation

When a plane cuts through a pyramid parallel to its base, the cross section is always the same shape as the base.
Since the base is a square, the cross section is also a square (a smaller, similar square).

Question 6 of 10

Michigan Standards 11A-11DMedium Calc Word Diagram

A swimming pool has the shape shown below — a rectangle with a semicircle on each end. Find the total area of the pool. (Use π ≈ 3.14)

A200.0 m²

B257.0 m²

C356.0 m²

D278.5 m²

Explanation

Rectangle area = 20 × 10 = 200 m².
Two semicircles = one full circle with r = 5: π × 5² = 3.14 × 25 = 78.5 m².
Total = 200 + 78.5 = 278.5 m².

Question 7 of 10

Michigan Standards 13A-13EMedium Calc Word Diagram

A dart is thrown randomly at the square board below. The board has a side length of 20 cm and contains a shaded circle with a radius of 8 cm. What is the probability that the dart lands inside the shaded circle? (Use π ≈ 3.14)

A62.8%

B40.0%

C50.2%

D78.5%

Explanation

Area of square = 20² = 400 cm².
Area of circle = πr² = 3.14 × 8² = 3.14 × 64 = 200.96 cm².
P = circle/square = 200.96/400 ≈ 0.5024 ≈ 50.2%.

Question 8 of 10

Michigan Standards 13A-13EMedium Calc Word Diagram

The two-way table shows student preferences. If a student is randomly selected from those who prefer Math, what is the probability they are in 10th grade?

A20%

B30%

C60%

D40%

Explanation

📌 Step 1: Identify the condition
"Given that a student prefers Math" means we only look at the Math column.

📌 Step 2: Find the relevant numbers
Total who prefer Math = 30
10th graders who prefer Math = 12

📌 Step 3: Calculate conditional probability
P(10th | Math) = 12/30 = 2/5 = 0.40 = 40%

💡 Key concept: Conditional probability restricts the sample space. We're not looking at all 60 students — only the 30 who prefer Math.

Question 9 of 10

Michigan Standards 12A-12EMedium Calc Word Diagram

A tangent line touches circle O at point T. OT = 5 and the external point P is 13 units from the center O. What is the length of tangent segment PT?

A12

C14

D10

Explanation

The tangent is perpendicular to the radius at the point of tangency. Using the Pythagorean theorem: PT = √(OP² − OT²) = √(13² − 5²) = √(169 − 25) = √144 = 12.

Question 10 of 10

Michigan Standards 3A-3DEasy Calc Word Diagram

Which of the following figures has BOTH reflectional and rotational symmetry?

AC (Parallelogram)

BB (Regular hexagon)

CD (Arrow)

DA (Scalene triangle)

Explanation

📌 Step 1: Check each figure

A (Scalene triangle): No lines of symmetry, no rotational symmetry ✗
B (Regular hexagon): 6 lines of symmetry + rotational symmetry at 60° ✓
C (Parallelogram): No lines of symmetry, rotational symmetry at 180° only → partial ✗
D (Arrow): 1 line of symmetry (vertical) but no rotational symmetry ✗

📌 Answer: B (Regular hexagon)

💡 Tip: All regular polygons have BOTH reflectional AND rotational symmetry. The number of symmetry lines = number of sides.

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