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Introduction to Triangles

A triangle is one of the most fundamental shapes in geometry, forming the basis of numerous mathematical concepts and real-world applications. It is a polygon with three sides, three vertices, and three angles, with the sum of its interior angles always equal to \boldsymbol{180^\circ}.

Triangles are widely classified based on their sides and angles.

Classification of Triangles

By Sides

  1. Equilateral Triangle: All three sides are equal, and each angle measures \boldsymbol{60^\circ}.
  2. Isosceles Triangle: Two sides are equal, and the angles opposite these sides are also equal.
  3. Scalene Triangle: All sides and angles are unequal.

By Angles

  1. Acute Triangle: All angles are less than \boldsymbol{90^\circ}.
  2. Right Triangle: One angle measures exactly \boldsymbol{90^\circ}.
  3. Obtuse Triangle: One angle is greater than \boldsymbol{90^\circ}.

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. This is a cornerstone of geometry, used extensively in solving problems related to height, distance, and measurement.

Criteria for Similarity

  1. AA (Angle-Angle) Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
  2. SSS (Side-Side-Side) Criterion: If the corresponding sides of two triangles are in proportion, the triangles are similar.
  3. SAS (Side-Angle-Side) Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, the triangles are similar.

Example

Problem: In \boldsymbol{\triangle ABC} and \boldsymbol{\triangle PQR}, \boldsymbol{\angle A = \angle P}, \boldsymbol{\angle B = \angle Q}, and \displaystyle\boldsymbol{\frac{AB}{PQ} = \frac{AC}{PR}}. Show that \boldsymbol{\triangle ABC \sim \triangle PQR}.

Solution:
By the AA criterion:

\boldsymbol{\triangle ABC \sim \triangle PQR \text{ since } \angle A = \angle P \text{ and } \angle B = \angle Q.}

Additionally, \displaystyle\boldsymbol{\frac{AB}{PQ} = \frac{AC}{PR}} satisfies the SAS criterion.

Important Theorems on Triangles

  1. Basic Proportionality Theorem (Thales’ Theorem):
    If a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those sides in the same ratio:
    \displaystyle\boldsymbol{\frac{AD}{DB} = \frac{AE}{EC}}
  2. Pythagoras Theorem:
    In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
    \boldsymbol{c^2 = a^2 + b^2}

Example Using Pythagoras Theorem

Problem: In a right-angled triangle, the lengths of two sides are \boldsymbol{6 , \text{cm}} and \boldsymbol{8 , \text{cm}}. Find the hypotenuse.

Solution:
Using the formula:

\boldsymbol{c^2 = a^2 + b^2}

Substituting the values:

\boldsymbol{c^2 = 6^2 + 8^2} \boldsymbol{c^2 = 36 + 64} \boldsymbol{c^2 = 100} \boldsymbol{c = \sqrt{100} = 10}

Answer: \boldsymbol{c = 10 , \text{cm}}

Applications of Similarity in Triangles

  1. Measuring Heights and Distances:
    Triangle similarity is used in indirect measurement techniques, such as calculating the height of a building using shadows.
  2. Map Scaling:
    Similar triangles are used in maps and scale models to represent real-world distances.

Practice Problems

  1. Problem: In \boldsymbol{\triangle ABC}, \boldsymbol{DE \parallel BC}, and \boldsymbol{AD : DB = 3 : 5}. Find \displaystyle\boldsymbol{\frac{AE}{EC}}.
    Solution:
    Using the Basic Proportionality Theorem:
    \displaystyle\boldsymbol{\frac{AE}{EC} = \frac{AD}{DB} = \frac{3}{5}}
  2. Problem: In a triangle, the sides are \boldsymbol{7 , \text{cm}, 24 , \text{cm}, \text{and } 25 , \text{cm}}. Show whether the triangle is right-angled.
    Solution:
    Using Pythagoras Theorem:
    \boldsymbol{c^2 = a^2 + b^2}
    Substituting the values:
    \boldsymbol{25^2 = 7^2 + 24^2}
    \boldsymbol{625 = 49 + 576}
    \boldsymbol{625 = 625}
    The triangle satisfies Pythagoras Theorem, so it is a right-angled triangle.

FAQs

How do you prove that two triangles are similar?2024-12-18T14:23:04+05:30

To prove similarity, use the AA, SSS, or SAS criterion:

    • \boldsymbol{\text{AA: Two angles are equal}}.
    • \boldsymbol{\text{SSS: All sides are proportional}}.
    • \boldsymbol{\text{SAS: Two sides are proportional, and the included angle is equal}}.
What are the real-life applications of similar triangles?2024-12-18T14:22:03+05:30

Similar triangles are used in:

    • Indirect measurement techniques (e.g., finding heights of buildings).
    • Map scaling and architectural designs.
What are the properties of a right-angled triangle?2024-12-18T14:21:31+05:30

In a right-angled triangle, one angle measures \boldsymbol{90^\circ}, and the Pythagoras Theorem holds: \boldsymbol{c^2 = a^2 + b^2}.

What is the sum of the interior angles of a triangle?2024-12-18T14:19:00+05:30

The sum of the interior angles of a triangle is always \boldsymbol{180^\circ}.

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