Geometry Theorems — Medians, Angle Bisectors, and Altitudes in a Triangle
Geometry Theorems — Medians, Angle Bisectors, and Altitudes in a Triangle. Practice questions to deepen understanding of medians, angle bisectors, and altitudes in a triangle. Online math practice with full solutions and step-by-step explanations.
Medians, angle bisectors, and altitudes practice — centroid, incenter, orthocenter, Euler line. Special points in a triangle.
Median (vertex → midpoint of opposite side), angle bisector, altitude.
📏 Definition:
A median in a triangle is a segment connecting:
Median in a triangle = a segment connecting a vertex to the midpoint of the opposite side
If M is the midpoint of BC,
then AM is a median from vertex A ✓
There are three medians (one from each vertex) ✓
⊙ Intersection point:
The three medians of a triangle:
The three medians of a triangle meet at one point!
This point is called: centroid ✓
Notation: G
📐 Division:
The centroid divides every median in the ratio:
The centroid divides every median in the ratio 2:1
The part from the vertex to the centroid = twice the part from the centroid to the base
AG : GM = 2 : 1 ✓
If the median is 9 cm long:
• From vertex to centroid: 6 cm
• From centroid to base: 3 cm
6:3 = 2:1 ✓
∠ Definition:
An angle bisector in a triangle is a ray that:
Angle bisector = a ray emanating from a vertex and dividing the angle into two equal parts
If AD bisects ∠A,
then ∠BAD = ∠CAD ✓
There are three angle bisectors (one from each vertex) ✓
⊙ Intersection point:
The three angle bisectors of a triangle:
The three angle bisectors of a triangle meet at one point!
This point is called: incenter (center of the inscribed circle) ✓
Notation: I
The distance from I to each of the sides is equal!
This is the radius of the inscribed circle ✓
⊥ Definition:
An altitude in a triangle is a segment:
⊙ Point of intersection:
The three altitudes of a triangle:
🔢 Calculation:
The median in a triangle is 12 cm long. The distance from the vertex to the centroid is:
Length of median AM = 12 cm
AG = ? (from the vertex to the centroid)
Solution:
The centroid divides the median in a 2:1 ratio
The median is divided into 3 equal parts:
AG = (2/3) × 12 = 8 cm ✓
GM = (1/3) × 12 = 4 cm
8:4 = 2:1 ✓
📐 Theorem:
An angle bisector in a triangle divides the opposite side:
⊿ Special case:
The median to the hypotenuse in a right triangle is equal to:
△ Special case:
In an equilateral triangle, the altitude is also:
⊙ Inscribed circle:
An inscribed circle in a triangle is a circle:
⊙ Circumscribed circle:
A circumscribed circle of a triangle is a circle:
🔢 Calculation:
In triangle ABC, AB=6, AC=9. The angle bisector from A meets BC at point D. If BD=4, then DC equals:
⊥ Perpendicular bisector:
A perpendicular bisector of a segment is a line:
⊙ Intersection point:
The three perpendicular bisectors in a triangle meet at:
The three perpendicular bisectors of the triangle's sides meet at the center of the circumscribed circle!
Notation: O ✓
📐 Euler line:
The three points: centroid, orthocenter and circumcenter are:
The three points:
• O - circumcenter
• G - centroid
• H - orthocenter
lie on a single line!
This line is called: Euler line ✓
OG:GH = 1:2
The centroid divides the Euler line!
🔢 Computation:
In a right triangle, the legs are 6 and 8 cm. The altitude to the hypotenuse equals:
Leg a = 6 cm
Leg b = 8 cm
Altitude to hypotenuse h = ?
Solution:
1. Hypotenuse (Pythagorean theorem):
c² = 6² + 8² = 36 + 64 = 100
c = 10 cm
2. Area in two ways:
S = ½ab = ½ch
½×6×8 = ½×10×h
24 = 5h
h = 4.8 cm ✓
⊙ Special points:
In an equilateral triangle, all special points:
In an equilateral triangle, all special points coincide at one point!
• Centroid = G
• Incenter = I
• Circumcenter = O
• Orthocenter = H
G = I = O = H ✓
📚 Summary:
Which point is always inside the triangle?
Always inside the triangle! ✓
In every type of triangle
• Acute: inside
• Right: on the right angle
• Obtuse: outside the triangle!
• Acute: inside
• Right: at the midpoint of the hypotenuse
• Obtuse: outside the triangle!
Always inside the triangle! ✓