A triangle in geometry is a plane that is bounded by three straight lines. In the following figure PQR is a triangle. The triangle is expressed by ∆. Here we will try to master properties of 7 types of triangles.
Parts of Triangle in Geometry
A triangle has a total of three parts. And they are three arms, three angles and three vertices.
In the figure 1 PQ, QR and PR are the three arms of the triangle. ∠PQR, ∠QRP and ∠RPQ are the three angles and P, Q and R are the vertices of the triangle ∆PQR.
Properties of Triangle in Geometry
- The sum of the two smaller arms of the triangle is larger than the third arm.
- The sum of the three angles of a triangle is 180 degrees.
- The point at which the arms of a triangle meet is called the angular point.
- If one of the sides of a triangle imagines the ground, its opposite angular point is called the vertex.
- Any exterior angle of a triangle is equal to the sum of opposite interior angles.
Classification of Triangle in Geometry
There are six types of triangles in terms of arms and angles.
Triangles in terms of Arms
A triangle whose arms are equal in length is called an equilateral triangle.
In the figure 2, ∆ABC is and equilateral triangle where AB = BC = CA.
Properties of Equilateral triangle
- The length of each side of this triangle is equal.
- The value of each angle of this triangle is equal. And the angles are 60°.
A triangle whose two sides are equal to each other is called an isosceles triangle.
Here ∆MNO is an isosceles triangle where MN = MO and ∠MON = ∠MNO.
Properties of Isosceles triangle
- The lengths of the two sides of this triangle are equal to each other.
- The opposite angles of equal arms are equal.
A triangle whose three sides are unequal to each other is called a scalene triangle.
∆PQR is a scalene triangle. Here PQ ≠ QR ≠ PR.
Properties of Scalene Triangle
- The lengths of the three sides of the triangle are unequal to each other.
- The three angles are also unequal to each other.
Triangles in terms of Angles
A triangle whose one angle is a right angle is called a right-angled triangle.
∆PQR is a right-angled triangle. ∠PQR = 90° and PR is hypotenuse.
Properties of Right-angled Triangle
- The value of an angle is 90°.
- The sum of the other two angles is 90°.
- The length of the opposite side of its right angle is the largest; its name is the hypotenuse.
The triangle whose all the angles are acute angle is called acute triangle.
∆PQR is an acute triangle. Here ∠PQR, ∠QRP and ∠RPQ are all acute angle.
Properties of Acute Triangle
- The value of each angle is less than 90°.
An angle of a triangle which is obtuse, is called obtuse triangle.
∆XYZ is an obtuse triangle. Here ∠XYZ is obtuse i.e. greater than 90°.
Properties of Obtuse Triangle
- One of its angles is the obtuse angle.
- The other two angles are right angles.
- The length of the opposite arm of the obtuse angle is the largest.
Right-angled Isosceles triangle
A triangle in which one angle is right angled and the adjacent sides of the right angle are equal to each other is called a right-angled isosceles triangle.
In ∆PQR, ∠PQR = 90° and PQ = QR. So, ∆PQR is right-angled isosceles triangle.
Properties of Right-angled Isosceles triangle
- An angle of this triangle is a right angle.
- The value of each of the other two angles is 45°.
- The lengths of the two sides adjacent to the right angle are equal to each other.
Median of Triangle
The median is the straight line joining the midpoint of the opposite side from any angular point of a triangle.
In ∆PQR, PS is the median.
Properties of Median
- The triangle has three median.
- The medians meet at a point.
- The point at which the medians meet is called the centroid.
- The three median’s length of an equilateral triangle is equal.
The Height of a Triangle
The length of a perpendicular is called the height of the triangle if it is perpendicular to the opposite side of any triangle from the vertex to the opposite side or the extension of that side.
PN is the height of the triangle ∆PQR.
Properties of Height
- The height of a triangle is the perpendicular distance from any of its angular points to the opposite side.
- The height can be located inside or outside the triangle.
Base of the Triangle
The side of a triangle that is parallel to the horizontal line is called the base of the triangle.
QR is the base of the triangle ∆PQR.
15 Types of Angle
Parallel Lines & Traversals
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