Heights and medians · Code with Martin

A height is a perpendicular from a vertex to the opposite side, a median joins a vertex to the midpoint of a side. We draw both.

Two important lines inside a triangle

You already know how to draw a triangle. Today you will draw two famous lines in it that you will need for areas and in physics.

A height is a segment from a vertex perpendicular to the opposite side (or its extension). Picture the side as a floor and from the vertex you drop a plumb line — a stone on a string. Where it lands, the height ends. Every triangle has three heights and they all meet at one point.

A median is a segment from a vertex to the midpoint of the opposite side. No perpendicular — it just halves the side. There are three medians too and they meet at one point: the centroid. That is the point at which a triangle cut from cardboard you can hold on a fingertip — it is its centre of balance.

Height belongs to perpendicular, median to the midpoint of a side. Who does not mix those two lines up has won — and that mixing is what we work on today.

Adam cut a big triangle from cardboard for a school project and carries it on his palm — it keeps falling. Tereza: “Find the centroid.” They draw two medians: with a ruler they find the midpoints of two sides, join them to the opposite vertices. They mark the intersection with a dot. Adam supports the triangle with a finger exactly on the dot — and the cardboard does not move, it holds level. “Does it work with the third median too?” he asks. They draw it: it goes through the same point. “All three meet at one point. That is not luck, that is a theorem,” says Tereza.

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For a perpendicular use both marked edges of a set square: lay the long one on the side, the mark will lead perpendicular. A perpendicular by eye is the most common source of mistakes with heights.

How to draw a height and a median

Height from vertex C (labelled v by the side it falls on):

Step 1: lay a ruler with a mark so that the long edge lies on side AB (the opposite side). Step 2: slide the ruler along the side until the mark goes through vertex C. Step 3: draw a segment from C perpendicular to AB. The foot of the perpendicular (the landing point) is called the foot of the height.

Watch with an obtuse triangle: the height from a vertex at an acute angle falls outside the triangle — on the extension of the opposite side. Extend the side with a dashed line and drop the perpendicular onto the extension. It is not a mistake, it is a property of obtuse triangles.

Median from vertex C:

Step 1: find the midpoint of side AB — either measure and half, or more precisely with a compass (equal arcs from A and from B above and below the segment, the join of the intersections cuts AB at the midpoint). Step 2: join the midpoint to vertex C with a straight line.

Check theorems: three heights meet at one point (the orthocentre), three medians at the centroid. The centroid also splits each median in the ratio 2 : 1 — from the vertex it is twice as far as from the side. See, a ratio from lesson 12 in geometry.

A height in an acute triangle

A triangle with sides 7, 6 and 5 cm (draw by the SSS rule).

Height from vertex C to side AB: I lay a marked ruler with the long edge on AB, slide until the mark meets C, and draw the perpendicular. The foot of the height falls inside side AB.

I draw heights from A and from B the same way. All three lines meet at one point inside the triangle.

Check: with a protractor check that the height makes 90° with the side. An error over 2° means a slipped mark.

A height in an obtuse triangle falls outside

Draw a triangle with an angle of 120° at vertex C (sides say 4 and 5 cm by that angle).

Height from vertex A to side BC: I lay the ruler on BC — and I see that the perpendicular going through A meets only the extension of the side, not the side itself.

Method: I extend BC with a dashed line past vertex C, then drop a perpendicular from A onto the extension. The foot of the height sits outside the triangle.

The lesson: in an obtuse triangle two of the three heights fall outside. Draw the extension dashed, so it can be told from the sides.

The centroid of a cardboard triangle

Cut any triangle from cardboard.

Step 1: with a compass find the midpoints of two sides (arcs from both ends, join the intersections).

Step 2: draw two medians — join the midpoints to the opposite vertices.

Step 3: mark the intersection. Check with the third median: it must go through the same point.

Step 4: a physics check — support the triangle with a finger at the marked point. Does it hold level? You have the centroid. Measure the ratio too: from the vertex to the centroid it is twice as much as from the centroid to the side.

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A height is not a median. A height goes perpendicular (and need not hit the midpoint), a median goes to the midpoint (and need not be perpendicular). They coincide only in special triangles — for example an equilateral one. Before you draw a line, say out loud which of them you are drawing.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Drawing is on paper. Here calculate the angles of a triangle.

Practise the questions

On paper: lines inside a triangle

Draw with a sharp pencil, perpendiculars only with a mark. Label each line (v, t).

Draw a triangle with sides 8, 7 and 6 cm and construct all three heights. Mark their intersection.
In the same (or a new) triangle construct all three medians and mark the centroid. Measure in what ratio the centroid splits one median.
Draw an obtuse triangle and construct a height that falls outside the triangle. Draw the extension of the side dashed.
Draw an equilateral triangle with side 6 cm and check that the height from a vertex and the median from the same vertex coincide in one line.

Now you 💪

I know the definitions: height = perpendicular from a vertex to the opposite side, median = join of a vertex to the midpoint of the opposite side.
I know that in an obtuse triangle a height can fall on the extension of a side.
I find the centroid as the intersection of the medians and I know it splits the medians in the ratio 2 : 1.

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Done when: In your book you have a triangle with three heights, a triangle with medians and a centroid (ratio 2 : 1 measured) and an obtuse one with a height outside.

What to take from this lesson

Height: from a vertex perpendicular to the opposite side. Three heights, one intersection.
Median: from a vertex to the midpoint of the opposite side. Intersection = centroid.
The centroid splits a median in the ratio 2 : 1 from the vertex and is the centre of balance.
In an obtuse triangle heights fall on the extensions of the sides.
A perpendicular always with a mark, the midpoint of a side with a compass.