Triangles

Table of Contents
1. What is a Triangle?
2. Properties of a Triangle
3. Equilateral Triangle
4. Congruency
5. Mid-Point Theorem
6. Similar Triangles
7. Areas of Similar Triangles
8. Basic Proportionality Theorem
9. Angle Bisector Theorem
10. Intercept Theorem
11. Pythagoras Theorem
12. Other Theorems
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What is a Triangle?

• A triangle is a polygon with three sides.
• Triangles are classified in two general ways: by their sides and by their angles.

1. Classification Based on Sides

Based on sides, triangles have been classified into three categories:

1. Scalene Triangle: In this type of triangle, all three sides are of different lengths.

2. Isosceles Triangle: This triangle has two sides that are equal in length. The third side is called the base, and the angles opposite the equal sides are also equal.

3. Equilateral Triangle: All sides of this triangle are of the same length. It is also known as an equiangular triangle because all its angles are equal, each measuring 60°. This is because the sum of the interior angles in any triangle is 180°.

2. Classification Based on Angles

Triangles are also divided into three classes on the basis of the measure of the interior angles:

1. Obtuse Angled Triangle: In this triangle, one angle is greater than 90°. For example, in triangle ABC, if angle C is greater than 90°, then it is an obtuse-angled triangle.

2. Acute Angled Triangle: All angles in this triangle are less than 90°. For instance, triangle PQR is an acute-angled triangle because all its angles are under 90°.

3. Right Angled Triangle: This type of triangle has one angle that measures exactly 90°. In such a triangle, the side opposite the right angle is called the hypotenuse, while the other two sides are referred to as the legs, which are also known as the base and height. According to the properties of right-angled triangles, the relationship between the lengths of the sides is given by the Pythagorean theorem, which states that a² + b² = c², where c is the length of the hypotenuse.

Properties of a Triangle

1. The sum of the three angles in a triangle is always 180 degrees.

2. The sum of the exterior angles in any triangle is always 360 degrees, regardless of the type of triangle.

3. An exterior angle is equal to the sum of the two opposite interior angles.

4. The sum of the lengths of any two sides is always greater than the length of the third side.

5. The difference in length between any two sides is always less than the length of the third side.

6. The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

7. In a right-angled triangle, angle B is 90 degrees unless stated otherwise.

8. In a right-angled triangle whose angles are 30°, 60° and 90°:
- The side opposite to angle 30°=  Hypotenuse
- Side opposite to Angle 60°=  Hypotenuse.

9. Centroid:
(a) The point of intersection of the medians of a triangle. (Median is the line joining the vertex to the
mid-point of the opposite side).
(b) The centroid divides each median from the vertex in the ratio 2 : 1.
(c) To find the length of the median, we use the theorem of Apollonius.
AB² + AC² = 2(AD² + BD²)

(d) The medians will bisect the area of the triangle.
(e) If x, y, z are the lengths of the medians through A, B, C of a triangle ABC, then
"Four times the sum of the squares of medians is equal to three times the sum of the squares of the sides of the triangle".
4(x² + y² + z²) = 3(a² + b² + c²).

10. Orthocentre: This is the point of intersection of the altitudes. An altitude is a line drawn perpendicularly from a vertex to the opposite side. In a right-angled triangle, the orthocentre is at the vertex with the right angle.

11. Circumcentre: The point where the perpendicular bisectors of the sides intersect. 
(a) The circumcentre is the centre of the circle that passes through all the triangle's vertices. 
(b) The circumcentre is equidistant from the triangle's vertices. 
(c) If a, b, c are the sides of the triangle and Δ is the area, then abc = 4RΔ, where R is the radius of the circumcircle.

12. Incentre: This is the point of intersection of the internal bisectors of the angles of a triangle.


(c) Δ = rs if r is the radius of incircle, where s = semi-perimeter =
and Δ is the area of the triangle.
(d) BF = BD = s - b where 2s = a + b + c,
CE = CD = s - c
AF = AD = s - a
(e) The angle between the internal bisector and the external bisector is 90°.

NOTE:
30°, 60°, 90° triangle:

This is a special case of a right triangle whose angles are 30°, 60°, 90°.

In this triangle side opposite to angle 30° = Hyp/2.

Side opposite to Angle 60° = √3/2 x HYP.

Example 1. In ΔABC, AB = 9, BC = 10, AC = 12. Find the length of median through A.

• In the adjacent figure AD is the required median. Using
• Apollonius theorem in the triangle we have,
  2AD² + 2(5)² = 81+ 144 .
  2AD² + 50 = 225
  
  

Example 2. The sides of the triangle are 6 cm, 8 cm, and 10 cm. Find the area, Inradius and Circumradius of the triangle.

• s =  
  

Equilateral Triangle

In an equilateral triangle, all the sides are equal and all the angles are equal.
(a) Altitude =

(b) Area =

(c) Inradius =

(d) Circumradius =

NOTE:

(i) The equilateral Δ has maximum area for given the perimeter,

(ii) Of all the triangles that can be inscribed in a given circle, an equilateral triangle has maximum area.

Congruency

• Two or more shapes are considered congruent if they have the same size and shape. For flat shapes, this means that their corresponding sides and angles must be equal.
• Example: Triangles ABC and DEF are congruent if they are identical in every aspect (same shape and size). The symbols used for congruency are ≅ or ≡.

Congruent Triangles

• If ∠A =∠D, ∠B = ∠E, ∠C = ∠F
• AB = DE, BC = EF; AC = DF
• Then ΔABC ≡ ΔDEF or ΔABC ≅ ΔDEF

The Tests for Congruency:
(a) SAS Test:  This test checks if two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle. If they are, the triangles are congruent.
(b) SSS Test: According to this test, if all three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
(c) ASA Test: This test involves checking if two angles and one side of one triangle are equal to two angles and one side of another triangle. If they are, the triangles are congruent.
(d) RHS Test: The RHS test is specific to right-angled triangles. It states that if the hypotenuse and one side of one right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, then the two triangles are congruent.

Mid-Point Theorem

• A line segment that joins the midpoints of any two sides of a triangle is parallel to the third side and is half its length.
• In the adjacent triangle ABC, if D and E are the respective mid-points of sides AB & AC, then DE II BC and DE =   BC
  

Similar Triangles

• Two shapes are said to be similar when they have the same shape, regardless of their size. For two triangles to be considered similar, their corresponding angles must be equal, and the lengths of their corresponding sides must be in proportion.

• In the figure:
  ΔABC ∼ ΔDEF then,
  ∠A = ∠D, ∠B = ∠E & ∠C = ∠F
  
  

Test for Similarity of Triangles:

(a) AAA Similarity Test: This test states that if the three angles of one triangle are equal to the three angles of another triangle, then the two triangles are similar.

(b) SAS Similarity Test: According to this test, if the ratio of two corresponding sides of two triangles is equal and the angle between those sides is also equal, then the triangles are similar.

(c) SSS Similarity Test: This test states that if the ratios of all three corresponding sides of two triangles are equal, then the triangles are similar.

Areas of Similar Triangles

• The ratios of the areas of two similar triangles are equal to the ratio of the square of their corresponding sides, i.e., if ΔABC ∼ ΔDEF, then
  
• The ratio of the areas of two similar triangles is also equal to:
  (a) Ratio of the square of their corresponding medians.
  (b) Ratio of the square of their corresponding Altitudes.
  (c) Ratio of the square of their corresponding angle bisectors.
• If two triangles are similar, the following properties are true:
  (a) The ratio of the medians is equal to the ratio of the corresponding sides.
  (b) The ratio of the altitudes is equal to the ratio of the corresponding sides.
  (c) The ratio of the internal bisectors is equal to the ratio of the corresponding sides.

Basic Proportionality Theorem

• When a line is drawn parallel to one side of a triangle, it intersects the other two sides and divides them in proportion.

• If in ΔABC, DE is drawn parallel to BC, it would divide sides AB and AC proportionally, i.e.,
  
  
  

Important Result

• If in ΔABC DE II BC, and a line is drawn passing through A and parallel to BC.
  Then we will have:
  
  

Angle Bisector Theorem

• The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. i.e. In a Δ ABC in which AD is the bisector of ∠A, then
  
  

Intercept Theorem

• Intercepts made by two transversals (cutting lines) on three or more parallel lines are proportional. 
• In the figure, lines l and m are transversals to three parallel lines AB, CD, EF. Then, the intercepts (portions of lengths between two parallel lines) made, AC, BD & CE, DF, are respectively proportional.
  
  

Pythagoras Theorem

• The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
• For example, in triangle ABC, where angle C is the right angle, the theorem can be expressed as: AB² = AC² + BC ²
• Pythagorean Triplets are sets of three positive integers a, b, and c, such that a² + b² = c².
• These triplets represent the lengths of the sides of a right-angled triangle. Examples include: (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41), etc.
    

NOTE:

If you multiply the Pythagorean triplets by constant, then resultant will also be Pythagorean triplets e.g. (6, 8, 10), (18, 24, 30) etc.

Other Theorems

Acute Angled Theorem

• In an acute angle triangle ABC, AD is the altitude on BC from vertex A, and ∠ABC is the greatest angle among all the three angles. Then:
  

Obtuse Angled Theorem

• In an obtuse angle triangle ABC, AD is the altitude on CB produced from vertex A, and ∠ABC is the greatest angle among all three angles. 
• Then AC² = AB² + BC² + 2BD × BC
  

Apollonius' Theorem

• This theorem is the combination of the above two theorems and gives the length of the median. If in ΔABC, AD is the median, meeting side BC at D. Then