The Triangle and its Properties Class 7 Notes Maths Chapter 6

Students can go through AP 7th Class Maths Notes Chapter 6 The Triangle and its Properties to understand and remember the concepts easily.

Class 7 Maths Chapter 6 Notes The Triangle and its Properties

→ Triangle : A simple closed figure formed with three line segments.

A triangle PQR has

1. 3 sides namely PQ, QR and PR.
2. 3 angles namely ∠P, ∠Q and ∠R
3. 3 vertices namely P, Q, R.

→ In ΔPQR, the side opposite to vertex P is QR, the side opposite to vertex Q is PR and the side opposite to vertex R is PQ.

→ A triangle has 6 parts which are 3 angles and 3 sides.

→ There are three types of triangles based on sides.
i) Scalene Triangle : A triangle which no two sides are equal is called a scalene triangle.

In Δ ABC, AB ≠ BC ≠ AC.

ii) Isosceles Triangle: A triangle in which two sides are equal is called a Isosceles triangle.

In Δ PQR, PQ = PR.
Note : In an isoscoles triangle, the angles opposite to equal sides are equal.

In Δ ABC, AB = AC, then B = C

iii) Equilateral Triangle : A triangle in which all three sides are equal is called an equilateral triangle.

In Δ MNR, MN = MR = NR.
Note : In an equilateral triangle, all angles are equal and each is equal to 60°.

In Δ XYZ, XY = YZ = ZX, then
∠X = ∠Y = ∠Z = 60°

→ There are 3 types of triangles based on angles.
i) Acute angle triangle : In a triangle if all the angles are acute then it is called ah acute angle triangle.
Ex : 60°, 70°, 50°

ii) Obtuse angle triangle : A triangle in which one angle is greater than 90° i.e., obtuse then it is called obtuse angle triangle.
Ex : 100°, 50°, 30°

iii) Right angle triangle : A triangle in which one angle is right angle is called a right angle triangle.
Ex : 90°, 50°, 40°

→ The largest side of a right angle triangle lies exactly opposite to the right angle which is called as HYPOTENUSE

In ΔABC. ∠A = 90\ BC is the HYPOTENUSE

→ Right angle Isosceles triangle : A triangle in which are angle is right angle and other two angles are equal is called Right .Angled Isosceles

In Δ PQR. ∠Q = 90°, PQ – PR then
∠P = ∠R
∠P + ∠Q + ∠R = 180°
∠P + 90° + ∠P = 180°
2∠P = 180° – 90°
2∠P = 90°
∠P = \(\frac{90^{\circ}}{2}\) ⇒∠P = 45
∴ ∠P = ∠Q = 45°

→ Median of a Triangle : The line segment joining the mid point of one side of a triangle from the opposite vertex is called MEDIAN of the triangle.

In Δ ABC, D is the mid point of BC
AD is the median
BD = CD.

→ Number of medians of a triangle is 3.

→ Centroid : The point of intersection (or) concurrence of medians of a triangle is called CENTROID.

→ Centroid of a triangle divides each median in the ratio 2 : 1.

→ In Δ ABC, Q is the centroid. AD, BE and CF are the three medians.

AG : GD = 2 : 1
BG : GE = 2 : 1

→ A median connects a vertex of a triangle to the mid point of the opposite side.

→ Altitude of the triangle (or) Height of the triangle: In a triangle the perpendicular drawn from the vertex to the opposite side is called height (or) altitude of that triangle.

In Δ ABC, CD ⊥ AB and CD is the altitude (or) height.

→ There will be three altitudes (or) heights for a triangle.

In Δ ABC, AD ⊥ BC, BE ⊥ AC, CF ⊥ AB.

→ The altitudes of a triangle are concurrent i.e., they meet at one point.

→ Ortho centre : The point of intersection (or) concurrence of altitudes of a triangle is called ORTHOCENTRE.

→The altitude of an acute angle triangle always lies inside of the triangle.

→ The altitude of a right angled triangle is one of the sides containing right angle.

→ In case of obtuse angle triangle, the altitude lies outside if the base angle is obtuse.

→ In an isosceles triangle the median and altitude are same.

→ In an equilateral triangle the median and altitude are same.

In Δ ABC, AB = BC = AC, CD ⊥ AB and CD is the median and altitude.

→ Exterior angle of a triangle and its property.

In Δ ABC, ∠ACD is the exterior angle.

→ An exterior angle of a triangle is equal to the sum of its interior opposite angles.

Given : Consider Δ ABC.
∠ACD is an exterior angle.
To Show : m∠ACD = m∠A + m∠B
Through C draw \(\overline{\mathrm{CE}}\) , parallel to \(\overline{\mathrm{BA}}\).
Justification

The above relation between an exterior angle and its two interior opposite angles is referred to as the Exterior Angle Property of a triangle.
Ex : Find x in the figure.

x° = 50° + 70° ⇒ x° = 120°

→ Angle sum property of a triangle : The sum of the angles in a triangle is 180°.

In ΔPQR, ∠P + ∠Q + ∠R = 180°

→ The sum of angles of a triangle is 180° (or) 2 right angle.

→ If two angles of a triangle are given to find the third angle, we must subtract the sum of two given angles from 180°.
Ex : 1) If two angles of a triangle are 30° and 70°. Find the third angle.
Solution:
Third angle = 180° – (30° + 70°)
= 180° – 100°
= 80°

→ Prove that “The sum of the angles in a triangle is 180°.
Solution:

Given : In Δ ABC, ∠A, ∠B and ∠C are the three angles represented by ∠1, ∠2 and ∠3.
RTP : ∠1 + ∠2 + ∠3 = 180°
Construction : Draw l || BC, so that DE || CB.
Proof : l || BC, in Δ ABC, AB & AC are transversal, then ∠EAB = ∠2 (Alternate interior angles)
∠DAC = ∠3 (Alternate interior angles)
∴ ∠EAB + ∠BAC + ∠DAC = 180° (Straight line)
∠2 + ∠1 + ∠3 = 180°
∴ ∠1 + ∠2 + ∠3 = 180°

→ A triangle cannot have more than one obtuse angle,

→ A triangle cannot have two right angles.

→ A triangle can have 3 acute angles.

→ Sum of the length of two sides of a triangle

In Δ ABC, AB, BC AC are three sides
AB + BC > AC
AB + AC > BC ; BC + AC > AB
∴ The sum of any two sides of a triangle is greater than the third side.

→ To construct a triangle we need 3 independent measurements.

→ The difference of any two sides of a triangle is less than third side.

In ΔPQR, PQ, QR and PR are three sides
PQ – QR < PR
PR – QR < PQ ; PQ – PR < QR

→ Right angled triangle and Pythagoras property :

In the below triangle ∠B = 90°
AC is the side opposite to right angle and which is the longest side AC is called HYPOTENUSE.

→ In Δ PQR, if c² = a² + b², then ∠Q = 90°

→ In a square with side ’a’ units, then its diagonal = a√2 units.

Solution:
a² + a² = (a√2)²
2a² = a² × 2
2a² = 2a²

→ In a rectangle, the diagonal = \(\sqrt{l^2+b^2}\)

→ PYTHAGORAS PROPERTY : In a right angled triangle the sum of squares of two sides is equal to the square of hypotenuse.
Ex : In ΔABC, ∠C = 90°, then
AC² + BC² = AB²

→ The other two sides of a right angled triangle other than hypotenuse are called legs of the right angled triangle.

→ If the pythagoras property holds, the triangle must be right angled.

→ Pythagoras triplet : Take example of right angled triangle with measurements 3 cm, 4 cm and 5 cm respectively, then
5² = 25
4² = 16
3² = 9
5² = 4² + 3²
25 = 16 + 9 ⇒ 25 = 25
We called (3, 4, 5) as pythagoras triplet.

→ The other pythagoras triplets are
(12, 5, 13) (1, 1,√2 ), (40, 41, 9), (15, 17, 8), etc.