# Triangles

1.(a) In a triangle ABC, points A, B and C are called its vertices; the three angles are named as A, B and C or as BAC, ABC, ACB respectively. AB, BC, CD are its sides.

(b) Side AB is opposite to C, opposite BC is A, and opposite AC is B

â€¢ If two sides of a triangle are unequal, the longer side opposite to it.

â€¢ In a triangle, the greater angle has the longer side opposite to it.

2.(a) Sum of the angles of a triangle is 1800

(b) The longest side is opposite to the greatest angle and vice-versa. Also, the least side is opposite to the least angle. (c) Sum of any two sides is greater than the third side.

(d) If one angle of a triangle is greater than the side opposite to the second

(e) Exterior angle of a triangle is equal to the sum of the interior opposite angles.

b E In the above diagram, external angle E = sum of two internal angles a + b.

3.(a) A scalene triangle is one, which has all its sides unequal.

(b) An isosceles triangle has two of its sides equal.

(c) An equilateral triangle has all its sides equal.

4.(a) An obtuse triangle has one of its angles, as obtuse.

(b) An acute triangle has three acute angles.

(c) In a right triangle, one angle is a right angle.

5. There can be only one obtuse angle in a triangle.

6. There can be only one right angle in a triangle.

7.(i) Angles opposite to equal sides of a triangle are equal.

(ii) Sides opposite to equal angles of a triangle are equal.

# Congruence of two triangles

Two triangles are congruent if pairs of corresponding sides and corresponding angles are congruent. The symbol for congruence is â‰… There can be three cases: -

**
1. SSS case: -** If three sides of a triangle are congruent to corresponding sides of the other triangle, the two triangles are congruent.

**
2. SAS Case: -** If two sides and included angle of one triangle are respectively equal to the two sides and included angle of another triangle, the triangles are congruent.

**
3. ASA Case: -** If one side of a triangle and angles at the end points of the side are equal respectively to the corresponding side and angles of another triangle, the two triangles are congruent

**
4) RHS case:** - If two right triangles have one side and hypotenuse of one triangle equal respectively to the corresponding side and hypotenuse of the other triangle, the two triangles are congruent.

**Pythagoras Theorem:** In right triangles, the square of the hypotenuse equals the sum of the squares of the other two

It is important to learn the triplets that make right angled triangles. 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17.

9, 40, 41 11, 60, 61 12, 35, 37 16, 63, 65 20, 21, 29

These triplets satisfy the condition of AC2 = AB2 + BC2

If a, b, c denote the sides of a triangle then

(i) If c2 < a2 + b2, triangle is acute angled.

(ii) If c2 = a2 + b2, triangle is right angled.

(iii) If c2 > a2 + b2, triangle is obtuse angled.

**
Median:** A median of a triangle is the line from a vertex to the midpoint of the opposite side. The centroid is the point at which the medians of the triangle meet .The centroid divides the medians in the ratio 2:1. The median bisects the area of the triangles.

# Theorem of Appolonius

Sum of the squares of two sides of a triangle

= 2 (median) 2 + 2 (half the third side)2

In the figure AD is the median, then: AB2 + AC2 =2(AD)2 + 2(BD)2.

**Similar Triangles :-**

Two triangles are said to be similar if

i) their corresponding angles are equal

ii) their corresponding sides are proportional.

# Basic Proportionality Theorem

In a triangle a line drawn parallel to one side to intersect the other side in distinct points divides the two sides in the same ratio

- AAA Similarity: - If the corresponding angles are equal:
- SSS Similarity :- If the corresponding sides of two triangles are proportional.
- SAS similarity: If one pair of corresponding sides is proportional and the included angles are equal.

Midpoint theorem

In the above triangle, D and E are the midpoints of the sides AB and AC respectively. Then, DE is parallel to BC, and DE = 1/2 BC.

Special cases of triangles:-

**1. Equilateral triangle:** A triangle where all three sides are of equal length and each angle is 60^{o}.

Area: âˆš3/4 Ã— (side)Â²

Height = âˆš3/2 Ã— side

**2. Isosceles Triangle:** A triangle where opposite sides are of equal length.

Two angles are equal

Area: 1/2 base Ã— height.

# Important points with respect to a triangle

**Centroid:**** The point where the medians meet. **

The Centroid divides the each median in the ratio 2:1.

** **

**Orthocentre:**** The point where the perpendiculars from each vertex on the opposite side meet. **

**
Circumcentre:** The circumcentre is the point where the perpendicular bisectors of all the sides meet. A circle drawn with the circumcentre as the centre can circumscribe the triangle.

**
Incentre:** The incentre is the point where the three angle bisectors of the triangle meet. The inradius of the circle is the perpendicular distance from the incentre to any of the sides of the triangle. In the figure OD.

is the inradius. A circle drawn with O as is centre and radius DO can be inscribed in the triangle, with the sides forming the tangents to the circle. The incentre divides the bisector of any angle in the ratio of (b + c) : a.

Angle Bisector Theorem: If AD is the bisector of angle A,

Then, AB/AC = BD/DC

â˜≡Time Saver Relationship between area of triangle and circumradius: In the s formula (given above), where S = (a + b + c)/2, We have r Ã— S = abc/4R where r is the inradius and R is the circumradius of the triangle.

**Area of a triangle:**

There are 2 ways to find the area of a triangle:

(a) Area = 1/2 (base) (height)

(b) Area =s âˆš(s-a) (s-b)(s-c) where a, b, c are the sides of the triangle and s = (a+ b+ c)/2 and is called the semi perimeter.