Mathematics: Class 7 (Part 1) — Chapter 7: A TALE OF THREE INTERSECTING LINES

Equilateral triangles are the most symmetric ones, in which all the sides are of equal lengths.

Construction of Equilateral Triangles:

Constructing an equilateral triangle using just a marked ruler might require several trials. A more efficient method uses a compass:

1. Draw the base, say $AB$, of length $4$ cm.
2. Using a compass, construct a sufficiently long arc of radius $4$ cm from $A$.
3. Construct another arc of radius $4$ cm from $B$.
4. Let $C$ be the point of intersection of the arcs. Join $AC$ and $BC$ to get the required equilateral triangle.

To construct a triangle with given side lengths (e.g., $4$ cm, $5$ cm, and $6$ cm):

1. Construct the base $AB$ with one of the side lengths, say $4$ cm.
2. From $A$, construct a sufficiently long arc of radius $5$ cm.
3. From $B$, construct an arc of radius $6$ cm such that it intersects the first arc.
4. The point where both arcs meet is the required third vertex $C$. Join $AC$ and $BC$ to get $\triangle ABC$.

Types of Triangles based on side lengths:

• Equilateral triangles: all three sides are of equal length.
• Isosceles triangles: two sides are of equal length.
• Scalene triangles: three sides are of different lengths.

Not all sets of lengths can form a triangle. This is governed by the Triangle Inequality.

Concept of Triangle Inequality:

The direct straight-line path between any two points is shorter than any roundabout path via a third point.

If we consider a triangle with sides $a, b, c$, then for the triangle to exist, the following conditions must be met:

• $a + b > c$
• $a + c > b$
• $b + c > a$

If any of these conditions are not met, a triangle cannot exist (e.g., for lengths $10$ cm, $15$ cm, $30$ cm, $10 + 15 = 25$, which is not greater than $30$).

Visualising the construction of circles:

When constructing a triangle with sides $a, b, c$ (where $c$ is the longest side), we draw a base $AB$ of length $c$. Then, we draw circles (or arcs) with radii $a$ and $b$ centered at $A$ and $B$ respectively.

• If the sum of the two smaller lengths ($a+b$) is equal to the longest length ($c$), the circles touch each other at a point, but do not form a triangle.
• If the sum of the two smaller lengths ($a+b$) is less than the longest length ($c$), the circles do not intersect internally, and thus no triangle can be formed.
• If the sum of the two smaller lengths ($a+b$) is greater than the longest length ($c$), the circles intersect each other internally at two points, and a triangle can be formed.

Therefore, a triangle exists if and only if the triangle inequality is satisfied.

The sum of the three angles in any triangle is $180^\circ$. This is known as the angle sum property of triangles.

Proof using parallel lines:

Consider $\triangle ABC$. Draw a line $XY$ through vertex $A$ parallel to $BC$.

• $\angle XAB = \angle B$ (Alternate Interior Angles)
• $\angle YAC = \angle C$ (Alternate Interior Angles)
• Angles on the straight line $XY$ sum to $180^\circ$: $\angle XAB + \angle BAC + \angle YAC = 180^\circ$.
• Substituting the alternate interior angles: $\angle B + \angle BAC + \angle C = 180^\circ$.

This property can also be verified by folding a triangular cut-out of paper.

An exterior angle of a triangle is formed between the extension of a side and the other side (e.g., $\angle ACD$ in $\triangle ABC$ where $BC$ is extended to $D$).

The measure of an exterior angle of a triangle is equal to the sum of its two opposite interior angles.

Relation: $\angle ACD = \angle A + \angle B$.

Proof:

• $\angle ACB + \angle ACD = 180^\circ$ (Linear pair)
• $\angle A + \angle B + \angle ACB = 180^\circ$ (Angle Sum Property)
• From these two equations, $\angle ACD = 180^\circ - \angle ACB = (\angle A + \angle B + \angle ACB) - \angle ACB = \angle A + \angle B$.

The height or altitude of a triangle is the perpendicular line segment drawn from a vertex to its opposite side.

For $\triangle ABC$, if $AD$ is drawn perpendicular to $BC$, then the length of $AD$ is the height of vertex $A$ from $BC$. $AD$ is an altitude.

Construction of Altitudes using a set square and ruler:

1. Keep the ruler aligned to the base (e.g., $BC$).
2. Place the set square on the ruler such that one edge of its right angle touches the ruler.
3. Slide the set square along the ruler till its vertical edge touches the vertex (e.g., $A$).
4. Draw the altitude from the vertex to the base using the vertical edge of the set square.

Right-angled triangles (or right triangles) are triangles having one right angle. In such triangles, one of the sides can also be an altitude.

Triangles can be classified based on their side lengths or angle measures.

Classification based on side lengths:

• Equilateral triangles: All three sides are of equal length.
• Isosceles triangles: Two sides are of equal length.
• Scalene triangles: All three sides are of different lengths.

Classification based on angle measures:

• Right-angled triangles: Have one right angle ($90^\circ$).
• Obtuse-angled triangles: Have one obtuse angle (greater than $90^\circ$).
• Acute-angled triangles: All three angles are acute angles (less than $90^\circ$).

• Use of a compass simplifies the construction of triangles when the sidelengths are given.
• A set of three lengths satisfies the triangle inequality if the length of each side is smaller than the sum of the other two.
• A triangle can be constructed if and only if its sidelengths satisfy the triangle inequality.
• Triangles can be constructed when:
  • Two sides and their included angle are given.
  • Two angles and the included side are given.
• The sum of the angles of any triangle is always $180^\circ$.
• An altitude of a triangle is a perpendicular line segment from a vertex to its opposite side.
• Triangles are classified based on their angle measures as acute-angled, right-angled, and obtuse-angled triangles.