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Mathematics · Class 9 · Chapter 15: Projection of a Side of a Triangle

8 min

Key points

Projection of a side is the perpendicular distance from one vertex to the line containing the opposite side
In any triangle ABC, if D is the foot of perpendicular from B to AC, then AD is the projection of AB on AC
Projection can be positive or negative depending on the position of the foot of perpendicular
If the angle between two sides is acute, projection is positive; if obtuse, projection is negative
The formula for projection of side b on side a is: projection = b cos C
Projection is used to derive the cosine rule and solve various geometric problems
In right-angled triangles, one projection is always zero
Projection formula helps in finding unknown sides and angles in triangles

Definitions

Projection of a side: The projection of a side of a triangle on another side is the length of the perpendicular dropped from one end of the first side to the line containing the second side, measured from the foot of perpendicular to the other end of the second side.
Foot of perpendicular: The point where a perpendicular line from a vertex meets the opposite side (or its extension) is called the foot of perpendicular.
Positive projection: When the foot of perpendicular falls on the side itself (between the two vertices), the projection is said to be positive.
Negative projection: When the foot of perpendicular falls on the extension of the side (beyond one of the vertices), the projection is said to be negative.

Worked example

In triangle ABC, AB = 8 cm, BC = 6 cm and angle ABC = 60°. Find the projection of side BC on side AB. (4 marks)

The projection of side BC on side AB is 3 cm

Common mistakes

Students confuse which side's projection is being calculated and use wrong angle in formula
Forgetting to consider negative projection when angle is obtuse
Not drawing proper diagram showing foot of perpendicular clearly
Mixing up the cosine rule derivation with projection formula
Using wrong notation - writing projection of AB on BC instead of projection of AB on AC

Quick recap

Projection of a side of triangle is the foot of perpendicular drawn from the third vertex to that side or its extension. It can be calculated using cosine of the included angle. Projection is always measured along the line on which it falls, and represents the 'shadow' of one side on another.

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