Types of Triangles
Introduction
Triangles come in different types based on length of sides and angle measures. They're commonly used in construction for their stability. Knowing these properties helps solve real-world problems. Triangles are important because they offer a strong base, useful in building foundations and trusses. They're seen in traffic signals, pyramids, and even the Bermuda Triangle. There are six main types of triangles.
Understanding Triangles
Triangles are `2`- dimensional geometric shapes with three sides and three angles. When three lines intersect, they form a triangle. In geometry, a triangle is denoted by its vertices and sides. For example, in triangle `△ABC`, vertices are `A`, `B`, and `C`, and sides are `AB`, `BC`, and `AC`. One fundamental property of triangles is the angle sum property, which states that the total of all three interior angles of any triangle always adds up to `180^\circ`.
Basic Properties of a Triangle
- A triangle has three interior angles. The total of these angles inside any triangle is always `180^\circ`. This is called the angle-sum property.
- Another important rule is the triangle inequality property, which states that the sum of the lengths of two smaller sides of a triangle is always greater than the length of the longest side. For example, in a triangle with side lengths of `5`, `7`, and `10` units, the sum of the lengths of the two smaller sides (`5` and `7`) is `12`, which is greater than the length of the longest side `(10)`.
- In any triangle, the longest side is always opposite to the largest angle, while the shortest side is opposite to the smallest angle.
Different Types of Triangles
In math, triangles are `2`-dimensional shapes with three sides. They come in various types, mainly determined by their sides and angles. There are `3` types of triangles based on side lengths:
- Scalene
- Isosceles
- Equilateral
Additionally, triangles are classified by their angles:
- Acute triangle if all angles are less than `90°`
- Right triangle if one angle is exactly `90°`
- Obtuse triangle if one angle is greater than `90°`
Types of Triangles Based on Sides
We can classify types of triangles by sides. Triangles can be classified based on their side lengths into three triangle types:
`1`. Equilateral Triangle: An equilateral triangle is a type of triangle where all three sides are of equal length. This means that each angle in an equilateral triangle measures `60^\circ`.
`2`. Isosceles Triangle: An isosceles triangle is a triangle that has at least two sides of equal length. Consequently, the angles opposite to these equal sides are also equal.
`3`. Scalene Triangle: A scalene triangle is a triangle in which all three sides have different lengths. As a result, the measure of the three interior angles of a scalene triangle are also different from each other.
Types of Triangles Based on Angles
Triangles can be classified based on their interior angles into three types:
`1`. Acute Triangle: An acute triangle is a type of triangle where all three angles are acute angles, meaning each angle measures less than `90^\circ`.
`2`. Right Triangle: Right triangles are characterized by having one angle that measures exactly `90^\circ`, which forms a right angle. The side opposite to this right angle is known as the hypotenuse, and it's the longest side in a right triangle.
`3`. Obtuse Triangle: Obtuse triangles have one angle that is greater than `90^\circ`, known as an obtuse angle. This means that one of the triangle's corners is wider than a right angle.
Types of Triangles Based on Sides and Angles
There are various other types of triangles based on their sides and angles. Some of them are as follows:
`1`. Equiangular Triangle: A triangle in which all three sides and angles are the same length and measure.
`2`. Right Isosceles Triangle: A triangle which has two sides of equal length and one angle that measures exactly `90` degrees.
`3`. Obtuse Isosceles Triangle: A triangle having two sides of equal length and one angle that measures more than `90` degrees.
`4`. Acute Isosceles Triangle: A triangle in which two sides are of equal length, and all three angles are acute, meaning they measure less than `90` degrees.
`5`. Right Scalene Triangle: A triangle in which one of the angles is a right angle (`90` degrees) and all three sides have different lengths.
`6`. Obtuse Scalene Triangle: A triangle in which one of the angles is obtuse (greater than `90` degrees) and all three sides have different lengths.
`7`. Acute Scalene Triangle: A triangle in which all three angles are acute (less than `90` degrees) and all three sides have different lengths.
Solved Examples
Example `1`: What type of triangle is it called if all the angles of a triangle are less than `90` degrees?
Solution:
If all the angles of a triangle are less than `90` degrees, the triangle is called an acute triangle.
Example `2`: In triangle ` \triangle ABC `, if angle ` \angle A ` measures ` 60^\circ ` and angle ` \angle B ` measures ` 40^\circ `, what is the measure of angle ` \angle C `?
Solution:
Using the angle sum property, we know that the sum of the interior angles of a triangle is ` 180^\circ `. Therefore, to find the measure of angle ` \angle C `, we subtract the sum of angles ` \angle A ` and ` \angle B ` from ` 180^\circ `:
`\angle C = 180^\circ - (\angle A + \angle B)`
`\angle C = 180^\circ - (60^\circ + 40^\circ)`
`\angle C = 180^\circ - 100^\circ`
`\angle C = 80^\circ`
Therefore, angle ` \angle C ` measures ` 80^\circ `.
Example `3`: If the length of all the sides of a triangle are equal, what type of triangle is it?
Solution:
If the length of all the sides of a triangle are equal, the triangle can be identified as an equilateral triangle.
Example `4`: In a right triangle, if angle `X` measures ` 60^\circ `, find angle `Z`.
Solution:
Using the angle sum property:
`\angle X + \angle Y + \angle Z = 180^\circ`
`60^\circ + 90^\circ + \angle Z = 180^\circ`
`150^\circ + \angle Z = 180^\circ`
`\angle Z = 180^\circ - 150^\circ`
`\angle Z = 30^\circ`
Therefore, angle `Z` measures \(30^\circ \).
Example `5`: Is it possible to form a triangle with its sides measuring `5` cm, `7` cm and `9` cm?
Solution:
As per the properties of a triangle, sum of the lengths of two smaller sides of a triangle is always greater than the length of the longest side.
The two smaller sides are `5` cm and `7` cm.
Adding these sides we get `12` cm.
`12` cm is greater that the length of the longest side (`9` cm).
Hence, it is possible to form a triangle with its sides measuring `5` cm, `7` cm and `9` cm.
Practice Problems
Q`1`. In triangle `ABC`, if `AB = BC = 5` cm and `AC = 6` cm, what type of triangle is it?
- Equilateral
- Isosceles
- Scalene
- Right
Answer: b
Q`2`. Triangle `PQR` has angles measuring `50^\circ`, `35^\circ`, and `95^\circ`. What type of triangle is `PQR`?
- Equilateral
- Isosceles
- Scalene
- Acute
Answer: c
Q`3`. In triangle ` \triangle PQR `, if angle ` \angle P ` measures ` 45^\circ ` and angle ` \angle R ` measures ` 80^\circ `, what is the measure of angle ` \angle Q `?
- ` 110^\circ `
- ` 50^\circ `
- ` 55^\circ `
- ` 95^\circ `
Answer: c
Q`4`. If the length of all the sides of a triangle are different, what type of triangle is it?
- Scalene
- Isosceles
- Equilateral
- Right
Answer: a
Q`5`. Which set of measurements below is not capable of forming the sides of a scalene triangle?
- `6` cm, `8` cm, `10` cm
- `5` m, `12` m, `13` m
- `10` cm, `12` cm, `24` cm
- `17` m, `17` m, `18` m
Answer: c
Frequently Asked Questions
Q`1`. What is an equilateral triangle?
Answer: An equilateral triangle is a type of triangle in which all three sides are of equal length. Additionally, all three interior angles are also equal, each measuring `60` degrees.
Q`2`. How can you determine if a triangle is right, obtuse, or acute?
Answer: In a right triangle, one of the angles measures `90` degrees. In an acute triangle, all three angles are less than `90` degrees. In an obtuse triangle, one of the angles measures more than `90` degrees.
Q`3`. What is the sum of angles in a triangle?
Answer: The sum of the interior angles of a triangle is always `180` degrees. This property is known as the triangle angle sum theorem.
Q`4`. What is the difference between an isosceles and a scalene triangle?
Answer: An isosceles triangle has at least two sides of equal length, whereas a scalene triangle has all three sides of different lengths.
Q`5`. How do you find the perimeter of a triangle?
Answer: The perimeter of a triangle is found by adding the lengths of all three sides together.