NCERT Class 7 Ganita Prakash 1 Chapter 07 A Tale of Three Intersecting Lines PDF Download

Chapter 7: A Tale of Three Intersecting Lines

A triangle is the most basic closed shape. As we know, it consists of:

• three corner points, that we call the vertex of the triangle, and
• three line segments or the sides of the triangle that join the pairs of vertices.

Triangles come in various shapes. Some of them are shown below.

[Figure: Various triangles labeled ABC, YZX, WUV, and BAU, See in your textbook]

Observe the symbol used to denote a triangle and how the triangles are named using their vertices. While naming a triangle, the vertices can come in any order.

The three sides meeting at the corners give rise to three angles that we call the angles of the triangle. For example, in \( \triangle ABC \), these angles are \( \angle CAB \), \( \angle ABC \), \( \angle BCA \), which we simply denote as \( \angle A \), \( \angle B \) and \( \angle C \), respectively.

What happens when the three vertices lie on a straight line?

7.1 Equilateral Triangles

Among all the triangles, the equilateral triangles are the most symmetric ones. These are triangles in which all the sides are of equal lengths. Let us try constructing them.

Construct a triangle in which all the sides are of length 4 cm.

How did you construct this triangle and what tools did you use? Can this construction be done only using a marked ruler (and a pencil)?

Constructing this triangle using just a ruler is certainly possible. But this might require several trials. Say we draw the base - let us call it \( AB \) - of length 4 cm (see the figure below), and mark the third point \( C \) using a ruler such that \( AC = 4 \) cm. This may not lead to \( BC \) also having a length of 4 cm. If this happens, we will have to keep making attempts to mark \( C \) till we get \( BC \) to be 4 cm long.

[Figure: Triangle with base AB of 4 cm and point C above, showing AC = 4 cm and BC = 4 cm?, See in your textbook]

How do we make this construction more efficient?

Recall solving a similar problem in the previous year using a compass (in the Chapter 'Playing with Constructions'). We had to mark the top point of a 'house' which is 5 cm from two other points. The method we used to get that point can also be used here.

After constructing \( AB = 4 \) cm, we can do the following.

Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm from \( A \), as shown in the figure. The point \( C \) is somewhere on this arc. How do we mark it?

[Figure: Line segment AB of 4 cm with an arc of radius 4 cm centered at A, See in your textbook]

Teacher's Note

The key idea here is that a compass can draw a circle of any fixed radius. All points on the arc from \( A \) are exactly 4 cm away from \( A \). Similarly, all points on an arc from \( B \) with the same radius are 4 cm from \( B \). Where these two arcs cross is the point that is 4 cm from both \( A \) and \( B \), which is exactly what you need for an equilateral triangle.

Step 2: Construct another arc of radius 4 cm from \( B \).

[Figure: Two overlapping arcs of radius 4 cm, one centered at A and one at B, intersecting at point C above the line segment AB, See in your textbook]

Let \( C \) be the point of intersection of the arcs.

The construction ensures that both \( AC \) and \( BC \) are of length 4 cm. Can you see why?

Step 3: Join \( AC \) and \( BC \) to get the required equilateral triangle.

[Figure: Equilateral triangle ABC with all sides 4 cm, constructed from two intersecting arcs, See in your textbook]

7.2 Constructing a Triangle When its Sides are Given

How do we construct triangles that are not equilateral?

Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.

As in the previous case, this triangle can also be constructed using just a marked ruler. But it will involve several trials.

Teacher's Note

When you have three different side lengths to construct, the method is almost the same as for equilateral triangles: draw one side, then use compass arcs from each endpoint with the required radii. The intersection point of those arcs gives you the third vertex. The only difference is that now the two arc radii are different (5 cm and 6 cm instead of both being 4 cm).

Key Points

• A triangle consists of three vertices (corner points) and three sides (line segments connecting the vertices).
• An equilateral triangle has all three sides of equal length and can be constructed efficiently using a compass by drawing arcs from two vertices.
• To construct a triangle with given side lengths, draw one side as the base, then use compass arcs from each endpoint with radii equal to the required side lengths; the arcs intersect at the third vertex.

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