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NCERT Solutions Ganita Prakash (Part 1) Chapter 7 –1697.4 Constructions Related to Altitudes of Triangles — In-text Questions

Book page 167 Updated on2026-09-05

Q1.
Consider a triangle ABC. What is the height of the vertex A from its opposite side BC, and how can it be measured?
Answer

Drop a perpendicular from A to BC and measure it. That perpendicular is the height.

Let AD ⟂ BC, with D on BC
Length of AD = height of A from BC
The segment AD is called an altitude of the triangle

The other two altitudes are BE and CF — the perpendiculars from B and from C to their own opposite sides.

Why it happens: "Height" always means the shortest distance, and the shortest distance from a point to a line is measured along the perpendicular. Any slanting segment from A to BC would be longer than AD, so it would not describe how high A really is.
Tip: Whenever the word height of a triangle is used, it means the altitude to whichever side is being taken as the base.
Q2.
What would the altitude from A to BC be in this triangle? (a triangle in which the angle at B is obtuse)
Answer

The foot of the perpendicular falls outside the triangle, so BC has to be extended first.

A B C D altitude
BC is extended beyond B, and the perpendicular from A meets that extension at D.
Extend BC beyond B
Drop the perpendicular from A to this extended line
Foot of the perpendicular = D
AD is the altitude from A to BC
Why it happens: The angle at B is obtuse, so A leans out past the end of the base. The line containing BC still runs under A — only the segment BC does not. Altitudes are measured to the line of the base, so the line is extended to receive the perpendicular.
Q3.
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.
Answer

Fold so that the two halves of the base lie exactly on each other, with the crease passing through the top vertex.

  1. Cut out ∆ABC and take BC as the base.
  2. Fold the paper so the crease passes through A and part of BC falls exactly on the rest of BC.
  3. Open it out. The crease from A to BC is the altitude.

Why the crease is perpendicular:

Folding places the base line on itself
The two angles that the crease makes with BC land exactly on each other
So the two angles are equal
They are also on a straight line, so together they make 180°
Each angle = 180° ÷ 2 = 90°
Why it happens: A fold is a mirror. If one part of a line reflects onto the other part, the crease must be its mirror line, and a mirror line always meets what it reflects at a right angle.
Q4.
Construct an arbitrary triangle. Label the vertices A, B, C taking BC to be the base. Construct the altitude from A to BC. Can you see how to do this?
Answer

Use a set square sliding along a ruler — a ruler alone cannot give an accurate 90°.

  1. Step 1: Keep the ruler aligned to the base BC. Place the set square on the ruler so that one of the edges of its right angle touches the ruler.
  2. Step 2: Slide the set square along the ruler till the vertical edge of the set square touches the vertex A.
  3. Step 3: Draw the altitude to BC through A using the vertical edge of the set square.
Why it happens: The ruler holds the direction of BC fixed. The set square carries a built-in right angle, so while it slides its vertical edge stays exactly perpendicular to BC. Sliding only moves the edge sideways until it reaches A — the direction never changes.
Tip: If the foot of the perpendicular falls outside the segment BC, simply extend BC with the ruler first and then slide.
Q5.
Does there exist a triangle in which a side is also an altitude? Visualise such a triangle and draw a rough diagram.
Answer

Yes — in a right-angled triangle. If ∠B = 90°, then AB is itself the altitude from A to BC.

A B C altitude = side AB
With ∠B = 90°, the side AB is already perpendicular to BC, so it is the altitude from A.
∠B = 90° → AB ⟂ BC
So the altitude from A to BC is AB itself
Similarly, CB is the altitude from C to AB

Triangles having one right angle are called right-angled triangles, or simply right triangles.

Why it happens: An altitude is just a perpendicular from a vertex to the opposite side. In a right-angled triangle the two arms of the right angle are already perpendicular to each other, so two of the three altitudes are sides of the triangle. Only the third altitude, from the right-angle vertex to the longest side, has to be drawn.
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