Week 10 · Parallel lines and medians: equal intercepts, the midsegment of a triangle and the medians of a triangle · 3 lessons
3
The medians of a triangle: meeting point, the 2 : 1 ratio and the median to the hypotenuse
Mathematics Prep 2 Term 1, Unit 3, Lessons 3–4
GoalFind lengths with the three medians of a triangle, which meet at one point and are cut 2 : 1, and use the median to the hypotenuse.
New words
median · the segment from a vertex to the midpoint of the opposite sidecentroid · the point where the three medians meet; the balance point of the triangleratio 2 : 1 · from the vertex, the centroid cuts a median into two parts, the first twice the secondaltitude · a different segment: from a vertex perpendicular to the opposite side
Explanation
A median joins a vertex to the midpoint of the opposite side. A triangle has three medians, and they always meet at one point G, the centroid. Cut a triangle from card and balance it on a pencil tip at G. The centroid cuts every median in the ratio 2 : 1, measured from the vertex. In the picture AG = 2 × GD, so AG is two thirds of the median AD and GD is one third. If a median is 15 cm, AG = 10 cm and GD = 5 cm. If you know the short part GD, the whole median is AD = 3 × GD. A median is not an altitude: it goes to the midpoint, and it is not necessarily perpendicular to the side. There is one more rule for right-angled triangles: the median from the right-angle vertex to the hypotenuse is half the hypotenuse. The midpoint of the hypotenuse is equally far from all three vertices. This works only for the median from the right angle, and only in a right-angled triangle.
Worked examples
In Δ ABC the medians AD, BE and CF meet at G. AD = 15 cm, so AG = 10 cm and GD = 5 cm. If BG = 12 cm, then GE = 6 cm and the median BE = 18 cm. If GF = 4 cm, then CG = 8 cm and CF = 12 cm.
In Δ ABC with a right angle at C, the legs are 6 cm and 8 cm. The hypotenuse is 10 cm. The median CM from the right angle to the midpoint M of the hypotenuse is half of it: CM = 5 cm. Notice that AM = MB = CM = 5 cm.
The median to the hypotenuse of a right-angled triangle is 7.5 cm and one leg is 9 cm. The hypotenuse is 2 × 7.5 = 15 cm. The other leg is √(15² − 9²) = √(225 − 81) = √144 = 12 cm. The perimeter is 9 + 12 + 15 = 36 cm.
Class activity
Balance a triangle. Cut a triangle from card. Fold each corner onto the opposite side to find the three midpoints, draw the three medians and mark where they meet. Balance the triangle on a pencil tip at that point. Then measure one median and check the 2 : 1 ratio.
Practice
1
The medians of Δ ABC meet at G. AD = 24 cm. Find AG and GD.
AG = 2/3 × 24 = 16 cm and GD = 8 cm.
2
G is the centroid and GE = 5 cm on the median BE. Find BG and BE.
BG = 10 cm and BE = 15 cm.
3
A right-angled triangle has hypotenuse 26 cm. How long is the median to the hypotenuse?
Half of 26: 13 cm.
4
The median from the right angle is 10 cm and one leg is 12 cm. Find the other leg.
Hypotenuse = 20; other leg = √(400 − 144) = √256 = 16 cm.
5
Sara says the median AD is perpendicular to BC in every triangle. Is she right?
No. A median goes to the midpoint; it is perpendicular only in special triangles such as an isosceles triangle with AB = AC.
