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10 FE practice problems: analytic geometry and roots, with solutions

By Dr. Mohamed Elansary, PhD Environmental EngineeringUpdated

These ten original problems practice analytic geometry and roots, a topic from the FE exam specification, using the Mathematics, Straight Line part of the FE Reference Handbook 10.6. Each problem gives the situation and the values with units. Work it with the handbook PDF open and commit to an answer before you open the solution. Every solution shows the handbook page, the equation, the substitution with units, a size check, and the mistake behind each wrong option, so a wrong pick tells you exactly what to fix.

How to use these problems

Give each problem an honest attempt before you open the solution: write the given values with units, find the equation in the FE Reference Handbook PDF, and commit to an answer. Then compare line by line. If you picked a wrong option, read the note for that option; each one names the mistake that produces it.

The problems

Problem 1 · FE, Analytic geometry and roots

Points P (10.0, 2.5) and Q (-1.0, 6.0) are given in ft. The straight-line distance between P and Q is most nearly:

  • A 11.5 ft
  • B 133 ft
  • C 14.5 ft
  • D 10.4 ft

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
A (11.5 ft)
Given
x1 = 10.0, y1 = 2.5, x2 = -1.0, y2 = 6.0
Find
distance PQ (ft)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = -1.0 - (10.0) = -11.0 ft; Δy = 6.0 - (2.5) = 3.5 ft
  2. d = √(Δx² + Δy²) = √(-11.0² + 3.5²) = √133.3 = 11.5 ft
Result
11.5 ft, 3 significant figures
Check
d is longer than |Δx| = 11.0 and |Δy| = 3.5 and shorter than their sum, 14.5.
Why the others are wrong
  • B: stopped before taking the square root
  • C: added the x and y differences instead of combining their squares
  • D: subtracted the squares instead of adding them

Problem 2 · FE, Analytic geometry and roots

Two sides of a triangular lot are b = 63.0 ft and c = 66.5 ft, and the angle between them is A = 148°. The third side a is most nearly:

  • A 124 ft
  • B 109 ft
  • C 128 ft
  • D 35.9 ft

Handbook: Mathematics, Law of Cosines, FE Reference Handbook 10.6

Show the worked solution
Answer
A (124 ft)
Given
b = 63.0 ft, c = 66.5 ft, A = 148°
Find
side a (ft)
Handbook
Mathematics, Trigonometry, Law of Cosines, page 39
Equation
a² = b² + c² - 2bc cos A
Substitute
  1. a² = b² + c² - 2bc cos A = (63.0)² + (66.5)² - 2(63.0)(66.5) cos 148° = 15,500 ft²
  2. a = √15,500 = 124 ft
Result
124 ft, 3 significant figures
Check
for A above 90°, a is longer than √(b² + c²) = 91.60 ft.
Why the others are wrong
  • B: left out the 2 in 2bc cos A
  • C: took the cosine with the calculator in radians
  • D: added the 2bc cos A term instead of subtracting it

Problem 3 · FE, Analytic geometry and roots

Points P (-12.5, 17.0) and Q (-6.5, -9.0) are given in ft. The straight-line distance between P and Q is most nearly:

  • A 25.3 ft
  • B 712 ft
  • C 20.6 ft
  • D 26.7 ft

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
D (26.7 ft)
Given
x1 = -12.5, y1 = 17.0, x2 = -6.5, y2 = -9.0
Find
distance PQ (ft)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = -6.5 - (-12.5) = 6.0 ft; Δy = -9.0 - (17.0) = -26.0 ft
  2. d = √(Δx² + Δy²) = √(6.0² + -26.0²) = √712.0 = 26.7 ft
Result
26.7 ft, 3 significant figures
Check
d is longer than |Δx| = 6.0 and |Δy| = 26.0 and shorter than their sum, 32.0.
Why the others are wrong
  • A: subtracted the squares instead of adding them
  • B: stopped before taking the square root
  • C: added the coordinates instead of subtracting them

Problem 4 · FE, Analytic geometry and roots

Two sides of a triangular lot are b = 87.5 ft and c = 38.0 ft, and the angle between them is A = 42°. The third side a is most nearly:

  • A 81.4 ft
  • B 64.5 ft
  • C 108 ft
  • D 118 ft

Handbook: Mathematics, Law of Cosines, FE Reference Handbook 10.6

Show the worked solution
Answer
B (64.5 ft)
Given
b = 87.5 ft, c = 38.0 ft, A = 42°
Find
side a (ft)
Handbook
Mathematics, Trigonometry, Law of Cosines, page 39
Equation
a² = b² + c² - 2bc cos A
Substitute
  1. a² = b² + c² - 2bc cos A = (87.5)² + (38.0)² - 2(87.5)(38.0) cos 42° = 4,158 ft²
  2. a = √4,158 = 64.5 ft
Result
64.5 ft, 3 significant figures
Check
for A below 90°, a is shorter than √(b² + c²) = 95.40 ft.
Why the others are wrong
  • A: left out the 2 in 2bc cos A
  • C: took the cosine with the calculator in radians
  • D: added the 2bc cos A term instead of subtracting it

Problem 5 · FE, Analytic geometry and roots

Two sides of a triangular lot are b = 89.5 m and c = 78.0 m, and the angle between them is A = 99°. The third side a is most nearly:

  • A 116 m
  • B 109 m
  • C 123 m
  • D 128 m

Handbook: Mathematics, Law of Cosines, FE Reference Handbook 10.6

Show the worked solution
Answer
D (128 m)
Given
b = 89.5 m, c = 78.0 m, A = 99°
Find
side a (m)
Handbook
Mathematics, Trigonometry, Law of Cosines, page 39
Equation
a² = b² + c² - 2bc cos A
Substitute
  1. a² = b² + c² - 2bc cos A = (89.5)² + (78.0)² - 2(89.5)(78.0) cos 99° = 16,280 m²
  2. a = √16,280 = 128 m
Result
128 m, 3 significant figures
Check
for A above 90°, a is longer than √(b² + c²) = 118.7 m.
Why the others are wrong
  • A: took the cosine with the calculator in radians
  • B: added the 2bc cos A term instead of subtracting it
  • C: left out the 2 in 2bc cos A

Problem 6 · FE, Analytic geometry and roots

Points P (-8.5, -18.5) and Q (4.5, 4.5) are given in m. The straight-line distance between P and Q is most nearly:

  • A 36.0 m
  • B 14.6 m
  • C 26.4 m
  • D 698 m

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
C (26.4 m)
Given
x1 = -8.5, y1 = -18.5, x2 = 4.5, y2 = 4.5
Find
distance PQ (m)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = 4.5 - (-8.5) = 13.0 m; Δy = 4.5 - (-18.5) = 23.0 m
  2. d = √(Δx² + Δy²) = √(13.0² + 23.0²) = √698.0 = 26.4 m
Result
26.4 m, 3 significant figures
Check
d is longer than |Δx| = 13.0 and |Δy| = 23.0 and shorter than their sum, 36.0.
Why the others are wrong
  • A: added the x and y differences instead of combining their squares
  • B: added the coordinates instead of subtracting them
  • D: stopped before taking the square root

Problem 7 · FE, Analytic geometry and roots

Points P (-9.0, 19.5) and Q (-16.5, 11.5) are given in m. The straight-line distance between P and Q is most nearly:

  • A 2.78 m
  • B 40.1 m
  • C 11.0 m
  • D 15.5 m

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
C (11.0 m)
Given
x1 = -9.0, y1 = 19.5, x2 = -16.5, y2 = 11.5
Find
distance PQ (m)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = -16.5 - (-9.0) = -7.5 m; Δy = 11.5 - (19.5) = -8.0 m
  2. d = √(Δx² + Δy²) = √(-7.5² + -8.0²) = √120.3 = 11.0 m
Result
11.0 m, 3 significant figures
Check
d is longer than |Δx| = 7.5 and |Δy| = 8.0 and shorter than their sum, 15.5.
Why the others are wrong
  • A: subtracted the squares instead of adding them
  • B: added the coordinates instead of subtracting them
  • D: added the x and y differences instead of combining their squares

Problem 8 · FE, Analytic geometry and roots

Points P (-2.5, 18.5) and Q (20.0, -14.5) are given in m. The straight-line distance between P and Q is most nearly:

  • A 39.9 m
  • B 1,600 m
  • C 55.5 m
  • D 24.1 m

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
A (39.9 m)
Given
x1 = -2.5, y1 = 18.5, x2 = 20.0, y2 = -14.5
Find
distance PQ (m)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = 20.0 - (-2.5) = 22.5 m; Δy = -14.5 - (18.5) = -33.0 m
  2. d = √(Δx² + Δy²) = √(22.5² + -33.0²) = √1,595 = 39.9 m
Result
39.9 m, 3 significant figures
Check
d is longer than |Δx| = 22.5 and |Δy| = 33.0 and shorter than their sum, 55.5.
Why the others are wrong
  • B: stopped before taking the square root
  • C: added the x and y differences instead of combining their squares
  • D: subtracted the squares instead of adding them

Problem 9 · FE, Analytic geometry and roots

Points P (15.5, 12.5) and Q (-14.5, -19.5) are given in m. The straight-line distance between P and Q is most nearly:

  • A 43.9 m
  • B 62.0 m
  • C 1,920 m
  • D 7.07 m

Handbook: Mathematics, Straight Line, FE Reference Handbook 10.6

Show the worked solution
Answer
A (43.9 m)
Given
x1 = 15.5, y1 = 12.5, x2 = -14.5, y2 = -19.5
Find
distance PQ (m)
Handbook
Mathematics, Straight Line (two points), page 37
Equation
d = √[(x2 - x1)² + (y2 - y1)²]
Substitute
  1. Δx = -14.5 - (15.5) = -30.0 m; Δy = -19.5 - (12.5) = -32.0 m
  2. d = √(Δx² + Δy²) = √(-30.0² + -32.0²) = √1,924 = 43.9 m
Result
43.9 m, 3 significant figures
Check
d is longer than |Δx| = 30.0 and |Δy| = 32.0 and shorter than their sum, 62.0.
Why the others are wrong
  • B: added the x and y differences instead of combining their squares
  • C: stopped before taking the square root
  • D: added the coordinates instead of subtracting them

Problem 10 · FE, Analytic geometry and roots

Two sides of a triangular lot are b = 19.0 ft and c = 46.5 ft, and the angle between them is A = 37°. The third side a is most nearly:

  • A 1,110 ft
  • B 33.3 ft
  • C 34.2 ft
  • D 62.7 ft

Handbook: Mathematics, Law of Cosines, FE Reference Handbook 10.6

Show the worked solution
Answer
B (33.3 ft)
Given
b = 19.0 ft, c = 46.5 ft, A = 37°
Find
side a (ft)
Handbook
Mathematics, Trigonometry, Law of Cosines, page 39
Equation
a² = b² + c² - 2bc cos A
Substitute
  1. a² = b² + c² - 2bc cos A = (19.0)² + (46.5)² - 2(19.0)(46.5) cos 37° = 1,112 ft²
  2. a = √1,112 = 33.3 ft
Result
33.3 ft, 3 significant figures
Check
for A below 90°, a is shorter than √(b² + c²) = 50.23 ft.
Why the others are wrong
  • A: stopped at a² (no square root)
  • C: took the cosine with the calculator in radians
  • D: added the 2bc cos A term instead of subtracting it

A new problem posts every day inside the lab, with the full worked solution the same evening.

Frequently asked questions

Where is analytic geometry and roots in the FE Reference Handbook?

Look in the Mathematics, Straight Line part of FE Reference Handbook 10.6. Each solution gives the exact page, so you can practice finding it in the PDF the way you will on exam day.

Are these real FE exam questions?

No. They are original problems generated from the handbook formulas and checked by code. Real exam questions are confidential, and sharing them breaks the NCEES agreement every examinee accepts.

How do I check my answer before opening the solution?

Check the units of your result and whether its size makes sense for the situation. Each solution ends with the same kind of check, so you can compare your habit with ours.

Sources

  1. NCEES FE Civil CBT exam specifications (PDF). Retrieved October 3, 2026.
  2. NCEES FE Environmental CBT exam specifications (PDF). Retrieved October 3, 2026.
  3. NCEES FE Mechanical CBT exam specifications (PDF). Retrieved October 3, 2026.
  4. NCEES FE Reference Handbook 10.6 (free PDF in MyNCEES). Retrieved October 3, 2026.