1. Median of a triangle
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Chapter 6
Triangle and its properties
Median of a triangle
Median of a triangle is a line segment joining a vertex to the midpoint of the opposing side, bisecting it.
A median connects a vertex of a triangle to the mid-point of the opposite side.
In the ∆ ABC, the line segment AD joining the mid-point of BC to its opposite vertex A is called a median of the triangle.
Properties of Median of a Triangle
Every triangle has exactly three medians one from each vertex and they all intersect each other at the triangle's centroid.
- The 3 medians always meet at a single point, no matter what the shape of the triangle is.
- The point where the 3 medians meet is called the centroid of the triangle. Point O is the centroid of the triangle ABC.
- Each median of a triangle divides the triangle into two smaller triangles which have equal area.
- In fact, the 3 medians divide the triangle into 6 smaller triangles of equal area.
In ∆ ABC, three medians are AD, CE and BF and they are intersecting the point O which is centroid of the triangle.
2. Altitude of a triangle
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Altitude of a triangle
An altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex).
In ∆ ABC, AD is the altitude of triangle ABC.
Through each vertex, an altitude can be drawn. So, there are at most three altitudes in a triangle.
Properties of Altitudes of a Triangle
- Every triangle has 3 altitudes, one from each vertex. AE, BF and CD are the 3 altitudes of the triangle ABC.
- The altitude is the shortest distance from the vertex to its opposite side.
- The 3 altitudes always meet at a single point, no matter what the shape of the triangle is.
- The point where the 3 altitudes meet is called the ortho-centre of the triangle. Point O is the ortho-centre of the triangle ABC.
- The altitude of a triangle may lie inside or outside the triangle.
3. Exterior angles of a triangle
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Exterior angles of a triangle
An exterior angle of a triangle is equal to the sum of the opposite interior angles.
In the above figure, ∠ACD is the exterior angle of the Δ ABC.
So, ∠ACD = ∠CAB + ∠CBA
At each vertex of a triangle, an exterior angle of the triangle may be formed by extending one side of the triangle.
4. Apply angle sum property of a triangle
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Apply angle sum property of a triangle
The sum of the measures of the three angles of a triangle is 180°
In Δ ABC,
∠A + ∠B + ∠C = 180°
5. Special triangles
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Special triangles
Two Special Triangles: Equilateral and Isosceles
A triangle in which all the three sides are of equal lengths and each angle has measure 60o is called an equilateral triangle.
In an equilateral triangle,
AB = BC = CA
And ÐA = ÐB = ÐC = 600
A triangle in which two sides are of equal lengths is called an isosceles triangle
In triangle XYZ,
XY = XZ
Sum of the Lengths of Two Sides of a Triangle
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In the above figure,
AB + BC > AC
Also, the difference between the lengths of any two sides of a triangle is smaller than the length of the third side.
6. Sides of a triangle
- Books Name
- class 7 Mathematics Book
- Publication
- ReginaTagebücher
- Course
- CBSE Class 7
- Subject
- Mathmatics
Sides of a triangle
Right-Angled Triangles and Pythagoras Property
In a right angle triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are known as the base and perpendicular of the right-angled triangle.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of base and perpendicular.
(Hypotenuse)2 = (Base)2 + (Perpendicular)2]
If a triangle holds Pythagoras property, then the triangle must be right-angled.
Problem 1: PQR is a triangle, right angled at P. If PQ = 10 cm and PR = 24 cm, find QR.
Solution:
Given: PQ = 10 cm, PR = 24 cm
Let QR be x cm.
In right angled triangle QPR,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2 [By Pythagoras theorem]
=> (QR)2 = (PQ)2 + (PR)2
=> x = 102 + 242
=> x = 100 + 576
=> x= 100 + 576 = 676
=> x = 676
=> x = √676
=> x = 26 cm
Thus, the length of QR is 26 cm.
Problem 2: A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.
Solution:
Let AC be the ladder and A be the window.
Given: AC = 15 m, AB = 12 m, CB = a m
In right angled triangle ACB,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2 [By Pythagoras theorem]
=> (AC)2 = (CB)2 + (AB)2
=> 152 = a2 + 122
=> 225 = a2 + 144
=> a2 = 225 – 144
=> a2 = 81
=> a = √81
=> a = 9 cm
Thus, the distance of the foot of the ladder from the wall is 9 m.