A Tale of Three Intersecting Lines | A

Question 2

Cut out a paper triangle. Fix one of the sides as the base. Fold it in such a way that the resulting crease is an altitude from the top vertex to the base. Justify why the crease formed should be perpendicular to the base.

Solution

Understand the Question

An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side (base).
Folding a straight edge (line segment) onto itself divides the straight angle ( $180^\circ$ ) at the fold into two equal angles of $90^\circ$ , creating a line perpendicular to the edge.
By adjusting the fold so that the crease passes through the top vertex, we obtain the altitude to the base.

Step 1 · Define Triangle and the Goal

Let the paper triangle be $\text{ABC}$ with side $\text{BC}$ as the base and vertex $\text{A}$ as the opposite (top) vertex. Diagram 1

An altitude is a line segment drawn from a vertex perpendicular to the opposite side. Thus, the fold line (crease) must:

Pass through vertex $\text{A}$ .
Be perpendicular to the base $\text{BC}$ .

Step 2 · Fold the Base Onto Itself and Justification

Diagram 2

Fold the paper so that the line segment along base $\text{BC}$ lies directly on top of itself.
When a straight line segment folds onto itself along a crease, the straight angle of $180^\circ$ along the edge is divided into two equal adjacent angles on either side of the crease: $\text{Angle on each side} = \dfrac{180^\circ}{2} = 90^\circ$
Hence, the crease line is perpendicular to $\text{BC}$ .

Step 3 · Align Crease Through Vertex A

While keeping the base $\text{BC}$ folded onto itself, slide the paper until the crease line passes exactly through vertex $\text{A}$ .
Press the paper to form the sharp crease.
Since this crease passes through vertex $\text{A}$ and meets base $\text{BC}$ at $90^\circ$ , it forms the altitude from vertex $\text{A}$ to base $\text{BC}$ .

Answer

The crease is perpendicular to the base because folding line segment $\text{BC}$ onto itself divides the straight angle ( $180^\circ$ ) into two equal adjacent angles of $90^\circ$ , creating the altitude through vertex $\text{A}$ .

Common Mistakes

Confusing Altitude with Median / Perpendicular Bisector: Folding vertex $\text{B}$ directly onto vertex $\text{C}$ creates the perpendicular bisector of $\text{BC}$ , which may not pass through vertex $\text{A}$ unless the triangle is isosceles or equilateral.
Not Keeping the Edge Aligned: If the edge $\text{BC}$ does not lie exactly on itself during the fold, the resulting crease will not meet the base at $90^\circ$ .

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