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Triangle Theorems Centroid Circumcenter Test Answer Key with Scoring – Geometry

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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers

Geometry Math Test Answer Keys – Mathbot, the Math Made Easy robot coach, with free printable test answer keys for Geometry
Looking for a triangle theorems centroid circumcenter test answer key? Then this answer key gives the solutions and the scoring guide of the Geometry test on Triangle Theorems and Triangle Centers. Each solution shows the theorem, the work and the scoring split. Proof answers show one valid chain of reasons; any logically complete proof earns the same credit.

Standards: G-CO.C.10, G-C.A.3 · Level: Grade 10 Geometry

Question 1 Isosceles and exterior angle (2 points)

  1. (180° − 56°) ÷ 2 = 62°.
  2. The exterior angle is supplementary to 62°: 180° − 62° = 118°. (Check: 56° + 62° = 118°.)

(a) 62°, (b) 118°

Scoring: 1 point per part.

Question 2 Exterior angle equation (3 points)

  1. 7x − 9 = (3x + 12) + (2x + 19) = 5x + 31, so 2x = 40 and x = 20.
  2. 7(20) − 9 = 131°.
  3. 3(20) + 12 = 72° and 2(20) + 19 = 59°; 72 + 59 = 131.

x = 20; 131°; 72° and 59°

Scoring: 1 point per part.

Question 3 Midsegment equation (2 points)

BC = 2MN: 7x + 9 = 10x − 6, so 15 = 3x and x = 5. Then BC = 7(5) + 9 = 44.

x = 5; BC = 44

Scoring: 1 point for x, 1 point for BC.

Question 4 Centroid (3 points)

  1. (−3 + 9 + 0) ÷ 3 = 2.
  2. (5 + 1 − 6) ÷ 3 = 0.
  3. The centroid is two thirds of the way from the vertex: ⅔ · 24 = 16.

Centroid (2, 0); 16 units

Scoring: 1 point per part.

Question 5 Circumscribed circle (3 points)

PQR
Solved figure: circumcenter (4, 3), radius 5.
  1. The legs lie on the axes, so the right angle is at P.
  2. The circumcenter is the midpoint of the hypotenuse QR: ((8 + 0) ÷ 2, (0 + 6) ÷ 2) = (4, 3).
  3. QR = √(8² + 6²) = 10, so the radius is 5.

Right angle at P; circumcenter (4, 3); radius 5

Scoring: 1 point per part.

Question 6 Prove a perpendicular (3 points)

Statement Reason
1. AB ≅ AC and D is the midpoint of BC Given
2. BD ≅ CD Definition of midpoint
3. AD ≅ AD Reflexive Property
4. △ABD ≅ △ACD SSS
5. ∠ADB ≅ ∠ADC CPCTC
6. ∠ADB and ∠ADC are a linear pair, so each is 90° Congruent angles in a linear pair are right angles
7. AD ⊥ BC Definition of perpendicular

Scoring: 1 point for the congruent triangles with reasons (steps 2 to 4), 1 point for step 5, 1 point for the right-angle conclusion.

Question 7 Inradius of a right triangle (2 points)

  1. (5 + 12 + 13) ÷ 2 = 15.
  2. Area = ½ · 5 · 12 = 30 and 30 = r · 15, so r = 2.

(a) 15, (b) r = 2

Scoring: 1 point per part.

Question 8 Garden planning (2 points)

Equal distance from the three corners (vertices) is the circumcenter, the intersection of the perpendicular bisectors. Equal distance from the three edges is the incenter, the intersection of the angle bisectors.

Spigot: circumcenter; bench: incenter

Scoring: 1 point for each correct center.

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