Geometry - Isosceles Triangle
- Introduction
- What Do You Mean by an Isosceles Triangle?
- Elements of Isosceles Triangle
- Properties of Isosceles Triangle
- Classification of Isosceles Triangle
- Perimeter of an Isosceles Triangle
- Area of an Isosceles Triangle
- Solved Examples on Isosceles Triangle
- Practice Problems
- Frequently Asked Questions
Introduction
Triangles are three-sided polygons in geometry that can be divided into three categories based on the length of their sides, such as:
- Equilateral triangle - `3` equal sides
- Isosceles triangle - Only `2` equal sides
- Scalene triangle - `3` unequal sides
The classification, properties, and formulas of an isosceles triangle will be covered in this article along with examples.
What Do You Mean by an Isosceles Triangle?
An isosceles triangle is a triangle that has two sides with the same length. Furthermore, the angles opposite to the two equal sides are also identical. In other words, “Any triangle having two congruent sides is said to be an isosceles triangle.”
Triangle `ABC` is an isosceles triangle as the sides `AB` and `AC` are equal. Also `∠B = ∠C`. The Isosceles triangle theorem states that “If a triangle's two sides are congruent, then the angles opposite to the equal sides are also congruent”.
Examples: We see several objects in the real world that have an isosceles triangular shape. Some real-world examples of isosceles triangles are:
Elements of an Isosceles Triangle
Legs: The two equal sides of an isosceles triangle form its legs. The two legs are represented by `AB` and `AC` in the triangle `ABC`.
Base: The third and unequal side of an isosceles triangle stands as the base of the triangle. The base in the triangle `ABC` is `BC`.
Vertex angle: The angle formed by the two equal sides of an isosceles triangle is known as the vertex angle. Hence, the vertex angle in the triangle `ABC` is `∠BAC`.
Base angles: Base angles are the angles that relate to the base of an isosceles triangle. The two base angles in the triangle `ABC` are `∠ABC` and `∠ACB`.
Properties of an Isosceles Triangle
Each geometric shape contains some specific properties that distinguish it from the others.
Some of the properties of isosceles triangles are listed below:
- The base of a triangle is the side that is not equal when a triangle has two sides that are identical.
- The two equal sides are always opposite equal-sized angles of the triangle.
- The altitude of an isosceles triangle is the perpendicular distance from its vertex to its base.
- The three angles of an isosceles triangle add up to `180^\circ`.
Classification of an Isosceles Triangle
Isosceles triangles are classified into three types:
Acute Isosceles Triangle: All three angles of the triangle must be smaller than 90°, and at least two of the angles must have the same measurements.
Example: `50°, 50°,` and `80°`
Right Isosceles Triangle: One angle must be equal to 90° and the other two of the angles must have equal measurements.
Example: `45°, 45°,` and `90°`
Obtuse Isosceles Triangle: One of the three angles is obtuse, but two of the acute angles have the same measurements.
Example: `30°, 30°,` and `120°`
Perimeter of an Isosceles Triangle
The sum of the equal side lengths and base length of a triangle is known as the perimeter of an isosceles triangle.
We can calculate its perimeter by the base `(b)` and sides `(a)` of an isosceles triangle. It is given by -
Perimeter `=` Equal Side Lengths `+` Base length
Perimeter `= ``a`` + ``a`` + ``b`
Perimeter `= 2``a`` + ``b`
Area of an Isosceles Triangle
The total area covered by an isosceles triangle is called its area.
We can calculate its area by the height `(h)` and the base `(b)` of the triangle. It is given by -
Area `= 1/2 \times` base ` \times `height
Solved Examples on Isosceles Triangle
Example `1`: What is the area of an isosceles triangle with a base of `8` cm and a height of `7` cm?
Solution:
Plug in the values of the base and height in the area formula.
Area `= 1/2 \times` base `\times` height
`= 1/2 × ``8`` × ``7`
`= 28` `\text{cm}^2`
Example `2`: An isosceles triangle has two sides of `5` cm and a base of `9` cm. Calculate the perimeter of the triangle.
Solution:
Plug in the values of the base and height in the perimeter formula.
Perimeter `= 2\times` side `+` base
`= 2 \times ``5`` + ``9`
`= 10 + 9`
`= 19` cm
Example `3`: An isosceles triangle has two sides of `10` cm and a perimeter of `34` cm. If the altitude of the triangle is `5` cm, calculate the area of the triangle.
Solution:
Plug in the values of the base and height in the perimeter formula.
Perimeter `= 2\times` side `+` base
`34= 2 \times ``10`` + ` base
`34= 20 + ` base
base `= 14` cm
Area `= 1/2 \times` base `\times` height
`= 1/2 × ``14`` × ``5`
`= 70` `\text{cm}^2`
Example `4`: An isosceles triangle has two sides of `10` m and an area of `50` `\text{m}^2`. If the altitude of the triangle is `5` m, calculate the perimeter of the triangle.
Solution:
Plug in the values of the base and height in the perimeter formula.
Area `= 1/2 \times` base `\times` height
`50= 1/2 × `base` × ``5`
base `= \frac{2 \times 50}{5}
`base `= 20` m
Perimeter `= 2\times` side `+` base
`= 2 \times ``10`` + ``5`
`= 20 + 9`
`= 25` m
Practice Problems
Q`1`. An isosceles triangle has an area of `12` sq. cm and a base of `8` cm. Calculate the height of the triangle.
- `12` cm
- `18` cm
- `3` cm
- `4` cm
Answer: c
Q`2`. For an isosceles triangle `ABC`, `∠A = ∠B`. Which of the following statements is true?
- `AB = AC`
- `AB = BC `
- `AC \ne BC`
- `AC = BC`
Answer: d
Q`3`. An isosceles triangle has a perimeter of `32` cm and a base of `6` cm. What is the measure of its congruent sides?
- `13` cm
- `17` cm
- `24` cm
- `30` cm
Answer: a
Frequently Asked Questions
Q`1`. What does an isosceles triangle mean?
Answer: Any triangle that has two equal sides is said to be isosceles. Furthermore, the angles opposite to the two equal sides are congruent.
Q`2`. What are the different types of isosceles triangles?
Answer: An isosceles triangle can be categorized as one of the following depending on the measures of the angles.
- Isosceles Acute triangle
- Isosceles Right triangle
- Isosceles Obtuse triangle
Q`3`. What are some real-life examples of isosceles triangles?
Answer: Isosceles triangles can be found in various real-life scenarios, including:
- Roof designs of houses and buildings, where the two equal sides represent the slopes of the roof.
- The design of bridges and arches, where the equal sides provide stability and support.
- The wings of airplanes, where the symmetry of the isosceles triangle helps distribute lift evenly.
- The shape of certain musical instruments, such as violins and guitars, where the body is often constructed using isosceles triangles to enhance resonance and acoustics.
- The design of flags, banners, and pennants, where isosceles triangles are frequently used to create symmetrical and visually appealing shapes.