[Maths Class Notes] on Types of Triangles Pdf for Exam

There are different types of triangles based on the sides and angles of triangles. A triangle can have all sides the same or all sides different. Similarly, a triangle can either have all angles the same or all angles different. The angles can be less than, more than or equal to 90degrees. Two triangles have the same angles but the different lengths of sides are called similar triangles. Thus, the ratio of the sides of the bigger and smaller triangle is always constant.

 

What is a Triangle?

By definition, a triangle is a two-dimensional shape having a flat surface that can be drawn on a piece of paper. A triangle consists of three sides, three vertices and three angles. In other words, we can say that a Triangle is the simplest polygon. Polygons are closed, flat, two-dimensional shapes having many corners. The minimum number of corners required to make a polygon is three because one or two corners cannot make a closed shape. Each corner makes up an angle. The most basic fact about the triangle is that the sum of all the interior angles of a triangle is always 180 degrees. 

 

Note: No angle should be zero degrees or 180 degrees as it would make a straight line rather than a triangle. Also, no triangle can have two angles more than 90 degrees as it won’t make a triangle.

 

Terms Related To Triangles:

A line segment joining a vertex of a triangle to the midpoint of the opposite side of the triangle is called a median. In figure ABC, D is the midpoint of AB. Thus AD forms the median of the triangle ABC. 

 

Similarly, a median can be drawn from the midpoint of BC as well as CA. In other words, a triangle can have three medians.

 

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The centroid of a triangle is the point of intersection of the three medians of a triangle.

Here, the midpoint of the medians AD, BE and FC is the centroid of the triangle ABC.

 

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The length of the perpendicular from a vertex to the opposite side of a triangle is called its altitude, and the side on which the perpendicular is drawn is called it’s base.

In the triangle, ABC, the perpendicular drawn to BC, that is AD is the altitude. The side BC is called the base of the triangle.

 

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The point of intersection (or concurrence) of the three altitudes of a triangle is called its orthocentre.

The meeting point (O) of the altitudes AD, CF and BE of the triangle is called the orthocentre.

 

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The point of intersection of internal bisectors of the angle of a triangle is called incentre. The incentre of a circle is also the center of the circle which touches all the sides of the triangle.

 

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The point of intersection of the perpendicular bisectors of the sides of a triangle ABC is called its circumcentre. 

 

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Circumcircle is the circle drawn keeping the circumcentre of the triangle as the center such that the circle passes through all the vertices of the triangle.

 

Types of Triangles

There are three types of triangles based on sides and based on angles. To work with different types of triangles it is important to know their properties. A triangle can be classified based on two factors:

Types of Triangles Based on Sides:

1. Scalene Triangle.

2. Isosceles Triangle.

3. Equilateral Triangle.

 

Scalene Triangles

Scalene triangles have all the sides non-congruent. Noncongruent is the term used for unequal sides of triangles. For example, if a triangle has sides of 12 cm, 30 cm, and 23 cm then it would be called a scalene triangle. Even a triangle with two sides the same cannot be called a scalene triangle. Like all the other triangles, even the sum of all the angles of the scalene triangle is exactly 180 degrees. The figure given below illustrates a scalene triangle.

 

Facts About the Scalene Triangle.

1. No sides and no angles of a scalene triangle are the same.

2. The side opposite the smallest angle is also the smallest. 

3. Its longest side is right opposite to its biggest angle.

4. It cannot be bisected into two halves. 

5. It has no line of symmetry.

6. A scalene triangle can be an acute scalene triangle, an obtuse scalene triangle or a right scalene triangle.

7. The area of a scalene triangle can be calculated by using Heron’s formula if all the sides are given.

8. When a scalene triangle is inscribed in a circle, each angle is half the angle subtended by the opposite side. 

9. The center of the circumscribing circle lies inside the triangle if all three angles are acute.

                    

Isosceles Triangles

An isosceles triangle is a triangle having two sides of equal length and one unequal side. Also, the angles opposite to the equal sides of an isosceles triangle are equal. Thus, we can say that an isosceles triangle has two equal sides as well as two equal angles. The equal sides are called legs and the unequal side is called the base. An isosceles triangle can be bisected into two equal right-angled triangles. 

The area of an isosceles triangle can be estimated:

1. If the measure of one angle and one side are given

2. If three sides of the triangle are given.

3. If two sides of the isosceles triangle and their included angle are given.

 

Facts About the Isosceles Triangle:

1. Two sides are congruent to each other.

2. The two angles opposite to the equal sides are congruent to each other. Thus, the angles are also congruent according to the isosceles triangle base angle theorem. 

3. Apex angle is the angle that is not congruent to the two congruent base angles.

4. The height drawn
   from the apex of an isosceles triangle divides the base into two equal parts and also divides the apex angle into two equal angles.

5. Area of Isosceles triangle = ½ × base × height

6. The perimeter of an Isosceles triangle = sum of all the three sides

7. The third unequal angle of an isosceles can be acute or obtuse.

8. The circumcenter of an isosceles triangle lies inside the triangle if all the three angles of the three triangles are acute. 

9. The sides of the triangle are the chords of the circumcircle.

10. If one of the angles is 90 degrees, then the circumcenter lies outside the triangle. 

11. Centroid is the intersection of the medians of the Isosceles triangle. 

12. The median drawn from Apex divides the triangle at right angles.

13. The perpendicular bisectors of an isosceles triangle intersect at its circumcenter.

14. The angle bisectors of an isosceles triangle intersect at the incenter.

15. The circle drawn with the incenter touches the three sides of the triangle internally.

16. Each median divides the isosceles triangle into two equal triangles having the same area.

17. Joining the midpoint of three sides divides the triangle into 4 smaller triangles of the same area.

18. When a circle with a diameter equal to the base is drawn:

1. For an obtuse-angled isosceles triangle, the apex lies inside the circle. 

2. For a right-angled isosceles triangle, the apex lies on the circumference. 

3. For an acute-angled isosceles triangle, the apex lies outside the triangle.

19. When the midpoint apex is taken as a radius and a circle is drawn with the midpoint of the base as the center 

1. For acute-angled isosceles, the base vertices lie inside the circle.

2. For right-angled isosceles, the base vertices lie on the circumference

3. For an obtuse-angled isosceles triangle, the base vertices lie outside the circle.

 

Equilateral Triangles

In an equilateral triangle all sides are equal with each interior angle of 60 degrees. An equilateral triangle is also called an equiangular triangle because all the angles are the same. The area of an equilateral triangle can be estimated in three cases: 

• If the measure of one angle and one side are given.

• If three sides of the triangle are given.

• If two sides of the triangle and their included angles are given.

 

Facts About Equilateral Triangle:

1. All the sides are the same.

2. All the angles of the equilateral triangle are the same.

3. The median, altitude, angle bisector, perpendicular bisector all coincide at one line.

4. The area of an equilateral triangle is [frac{sqrt{3}}{4}s^{2}]
   Here, s is the sides of an equilateral triangle.

5. The orthocenter, circumcenter, incenter and centroid all lie at the same point. 

6. Each altitude is a median of the equilateral triangle.

7. The centroid is the meeting point of the angle bisectors, medians as well as perpendicular bisectors of a triangle.

 

 

Types of Triangles based on Angles:

1. Acute Triangle.

2. Obtuse Triangle.

3. Right Triangle. 

 

1. Acute Triangles

All the interior angles of an equilateral triangle are less than 90 degrees such that the sum of all the angles is always 180 degrees. The figure given below illustrates an acute triangle.

 

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The figure given above is an acute triangle with all the interior angles less than 90 degrees (that is, 74, 50 and 56 degrees).  

 

Facts About Acute Triangles:

1. It has all three angles as acute.

2. Its perpendicular bisectors intersect at the circumcenter and median intersect at the centroid.

3. The circumcenter of an acute triangle lies inside the triangle.

4. The angle bisectors meet at the incenter of the circle. A circle can be drawn with the incenter of the triangle as the center of the circle to touch the three sides of the triangle internally.

5. Joining the midpoints of the three sides of the triangle results in 3 parallelograms having the same area and 4 triangles of t
   he same area.

 

2. Obtuse Triangles

One of the three interior angles of an obtuse triangle is greater than 90 degrees and the other two are less than 90 degrees such that the sum of all the interior angles is always 180 degrees. 

 

Facts About Obtuse Triangles:

1. It has two acute angles and one obtuse angle.

2. Its perpendicular bisectors intersect at the circumcenter.

3. The medians intersect at the centroid.

4. Its circumcenter always lies outside the triangle.

5. The angle bisectors meet at the incenter of the triangle. A circle is drawn with the incentre as its center touches the three sides of the triangle internally.

6. Medians split the triangle into two smaller triangles having the same area.

7. Joining the midpoints of the three sides of the triangle results in 3 parallelograms of the same area and 4 triangles of the same area.

 

3. Right-Angled Triangles

A right triangle is a triangle in which one of the angles is 90 degrees. The side opposite to the right angle (90-degree angle) is always the longest side called the hypotenuse and the side right next to the right angle is called perpendicular. The side on which the right triangle rests is called base or adjacent. The right isosceles triangle is a triangle having a perpendicular and base of the same length such that the angle between them is 90 degrees. The figure given below illustrates a right triangle.

 

Facts About Right-Angled Triangle

1. Out of the three angles of a right-angled triangle, one angle is greater than 90 degrees and the other two are acute angles.

2. The longest side of the right triangle is called the hypotenuse and the angle opposite to the hypotenuse is 90 degrees.

3. The area of a right triangle is half the product of the base and height.

4. The radius of the circumcircle is always half the hypotenuse and the center of the circumcircle is always the midpoint of the hypotenuse.

5. A line perpendicular to the hypotenuse from the right angle results in three similar triangles.