A Tale of Three Intersecting Lines · Lesson 8 of 10
Constructions Related to Altitudes of Triangles
“Understand perpendicular height and construct altitudes inside, outside, and along a triangle’s sides.”
• Define an altitude and distinguish the segment from its length. • Identify the base associated with a particular height. • Construct an altitude using a ruler and a set square. • Extend an opposite side when its altitude foot lies outside the side segment. • Recognise a side that acts as an altitude in a right triangle.
Height Depends on the Chosen Base
When we measure a tree’s height, we measure straight up from the ground rather than along a sloping branch. A triangle’s height uses the same perpendicular idea. Choose one side as the base, then measure the perpendicular distance from the opposite vertex to the line containing that base.
A perpendicular line segment from a vertex of a triangle to the line containing its opposite side. It may meet the side itself or an extension of that side.
Take BC as the base of ΔABC. Draw AD from A so that it meets the line BC at 90°. D is the foot of the perpendicular. AD is the altitude from A, and the length AD is the height for base BC. Saying “altitude” identifies a segment; saying “height” refers to its length.
The page’s vertical direction is not what defines an altitude. If the triangle is tilted, the altitude is tilted too, but it must still be perpendicular to the chosen base. A segment from the vertex that meets the base at some other angle is not an altitude.
Three Possible Bases, Three Altitudes
Any of the three sides can be chosen as the base. From A, draw the perpendicular to BC. From B, draw the perpendicular to AC. From C, draw the perpendicular to AB. These are the three altitudes of the triangle, although a particular drawing may emphasise only one.
Choosing a new base changes which altitude supplies its height. You should therefore name the base whenever a question about height might be unclear. For example, “the height to BC” means the length of the perpendicular from A, not the perpendicular from B or C.
When the Foot Lies Outside
In an obtuse triangle, some perpendiculars from vertices meet an extension of the opposite side rather than the side segment itself. This does not change the definition. Extend the opposite side as a straight line until it meets the perpendicular from the vertex.
Problem
AD is perpendicular to BC, with A the opposite vertex and D on the line BC. Which base has height AD?
- 1.Identify where the perpendicular starts: at vertex A.
- 2.The side opposite A is BC, so BC is the associated base.
- 3.AD is its altitude, and the length AD is its height. A sloping segment such as AB is not the height unless it is perpendicular to BC.
Altitudes Using Paper Folding
A paper fold can help you see why the altitude is perpendicular. Fold the triangular cut-out so that the crease passes through the selected vertex while the base line folds onto itself. If the foot lies on an extended base line, draw or imagine that extension when interpreting the fold.
At the crease, the two directions of the base line lie over one another when folded. The crease makes equal angles with those two directions. Together the angles form a straight angle, so each is 90°. This explains the perpendicular relationship instead of treating the fold as merely a convenient trick.
Construction of the Altitudes of a Triangle
A ruler helps draw a straight segment, but it does not by itself set a precise right angle. A set square supplies a 90° corner. Keeping it against a ruler lets you slide its perpendicular edge to the required vertex while maintaining the correct angle to the base.
- Draw and label a triangle ABC, taking BC as the base. Extend BC if the perpendicular foot will be outside the segment.
- Align a ruler with the line BC. Put one edge of the set square’s right angle along the ruler.
- Keep that edge against the ruler and slide the set square until its perpendicular edge passes through A.
- Draw from A along the perpendicular edge until it meets the base line. Label the foot D and show the right angle.
Problem
Construct BC = 5 cm, AB = 6 cm, and AC = 5 cm, then draw the altitude from A to BC.
- 1.Draw BC = 5 cm. From B draw an arc of radius 6 cm; from C draw an arc of radius 5 cm. Their intersection gives A. Join AB and AC.
- 2.Align the ruler with BC and slide a set square along it until the perpendicular edge passes through A.
- 3.Draw AD to BC and mark ∠ADB = 90°. AD is the required altitude; do not assume it passes through the midpoint of BC.
Problem
Construct ΔTRY with RY = 4 cm, TR = 7 cm, and ∠R = 140°, then draw the altitude from T to RY.
- 1.Draw RY = 4 cm and a 140° ray from R. Mark T at 7 cm from R on the ray, then join TY.
- 2.The vertex T is beyond R relative to the usual horizontal base RY. Its perpendicular does not meet the segment RY.
- 3.Extend the line RY beyond R and use the set square to drop a perpendicular from T to that extension.
- 4.The perpendicular is the altitude even though it lies outside the triangle.
A Side Can Also Be an Altitude
If ∠B is 90°, then AB is already perpendicular to BC. For base BC, the altitude from A is AB itself, with its foot at B. Similarly, BC is the altitude from C to base AB. The third altitude is drawn from the right-angle vertex B to the opposite side AC.
A triangle with one interior angle equal to 90°. It is also called a right triangle.
An altitude need not be inside the triangle, vertical on the page, or a line to the midpoint. It must start at the chosen opposite vertex and meet the line containing the base at 90°.
Quiz
What makes a segment an altitude to a chosen base?
AD is perpendicular to BC, with A opposite BC. What is AD’s associated base?
Where can an altitude foot lie in an obtuse triangle?
If ∠B = 90°, which segment is the altitude from A to BC?
Why is a set square useful when constructing an altitude?
Practice Problems
- Draw an acute triangle and construct all three altitudes with a ruler and set square. State the base for each.
- Construct BC = 5 cm, AB = 6 cm, AC = 5 cm and draw the altitude from A to BC.
- Construct ΔTRY with RY = 4 cm, TR = 7 cm, ∠R = 140°. Draw the altitude from T and explain why the base must be extended.
- Draw a right triangle with the right angle at B. Identify the two sides that act as altitudes and draw the third altitude.
- Tilt a triangle drawing and explain why its altitude to the selected base need not be vertical on the page.
- Make an altitude by paper folding. Explain why matching the base line onto itself creates two equal angles of 90°.
- A line from a vertex reaches the midpoint of the opposite side but is not perpendicular. Explain why it is not an altitude.
Key Takeaways
• An altitude joins a vertex perpendicularly to the line containing the opposite side. • Its length is the height for that selected base. • A triangle has an altitude corresponding to each of its three sides. • An altitude may meet an extension of a side and lie outside the triangle. • In a right triangle, the two sides meeting at the right angle can each act as an altitude.