Triangle Theorems Centroid Circumcenter Test – Geometry Math (Printable PDF)

Question 1 Isosceles and exterior angle (2 points)

An isosceles triangle has a vertex angle of 56°.

1. Find each base angle. (1 point)
2. Find the exterior angle at one base vertex. (1 point)

Question 2 Exterior angle equation (3 points)

In the figure, the exterior angle at C measures (7x − 9)°, and the remote interior angles measure (3x + 12)° and (2x + 19)°.

1. Find x. (1 point)
2. Find the exterior angle. (1 point)
3. Find both remote interior angles. (1 point)

Question 3 Midsegment equation (2 points)

MN is a midsegment of △ABC, parallel to BC. MN = 5x − 3 and BC = 7x + 9. Find x and BC.

Question 4 Centroid (3 points)

Triangle DEF has D(−3, 5), E(9, 1) and F(0, −6).

1. Find the x-coordinate of the centroid. (1 point)
2. Find the y-coordinate of the centroid. (1 point)
3. A median of a triangle is 24 units long. How far is the centroid from the vertex on that median? (1 point)

Question 5 Circumscribed circle (3 points)

Triangle PQR has P(0, 0), Q(8, 0) and R(0, 6).

1. Which vertex has the right angle? (1 point)
2. Find the circumcenter. (1 point)
3. Find the radius of the circumscribed circle. (1 point)

Question 6 Prove a perpendicular (3 points)

Given: AB ≅ AC, and D is the midpoint of BC. Prove: AD ⊥ BC.

Question 7 Inradius of a right triangle (2 points)

A right triangle has legs 5 and 12 and hypotenuse 13. Its area equals the inradius times the semiperimeter.

1. Find the semiperimeter. (1 point)
2. Find the radius of the inscribed circle. (1 point)

Question 8 Garden planning (2 points)

A triangular community garden has a water spigot that must be the same distance from all three corners, and a bench that must be the same distance from all three edges. Name the triangle center used for each.