Properties of Triangle and Height and Distance

Properties of Triangles

The plane figure formed by the joining of three lines is called triangle and whose sum of the angles is \[\angle A\] + \[\angle B\] + \[\angle C\] = 180⁰. A triangle is illustrated below:

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Some Properties of the Triangle are as Follows:

1. Sum of the measures of any two sides of the triangle is always greater than the measure of third side.

2. The side of the triangle that lies opposite to the greatest angle will be greatest in length and vice versa.

3. If a, b and c are the sides of a triangle and c is the largest side. Then,

1. If c² < a² + b² , the triangle is acute angled.

1. If c² = a² + b² , the triangle is right angled.

2. If c² > a² + b² , the triangle is obtuse angled.

4. Area of triangle = \[\frac{1}{2}\] х base х height 

5. Area of triangle = \[\sqrt{s(s-a)(s-b)(s-c)}\] where s = \[\frac{a+b+c}{2}\] and a, b and c being the length of sides.

Sine Rule

Let a, b and c be the sides opposite to angles A, B and C respectively, then,

\[\frac{a}{sin A}\] = \[\frac{b}{sin B}\] = \[\frac{c}{sin C}\]

Cosine Rule

Let a, b and c be the sides opposite to angles A, B and C respectively, then,

cos A = \[\frac{b^{2}+c^{2}-a^{2}}{2bc}\], cos B = \[\frac{c^{2}+a^{2}-b^{2}}{2ca}\], cos C = \[\frac{a^{2}+b^{2}-c^{2}}{2ab}\]

Important Terms of Triangle

1. Median and Centroid – A line joining the midpoint of a side of a triangle to the opposite vertex is called a median. In the given figure D, E, F are the midpoints of the sides BC, AC and AB respectively. Hence, AD, BE and CF are medians.

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A median cuts a triangle into two parts of equal area. The point where the three median of a triangle meet is called the centroid of the triangle. The centroid of triangle divides each median in the ratio 2:1, that is AG:GD = BG:GE = CG:GF = 2:1

2. Perpendicular Bisector and Circumcentre – A perpendicular bisector to any side is the line that is perpendicular to that side and is passing through its midpoint. The point of intersection of three perpendicular bisectors of the triangle is called circumcentre. The circumcentre is equidistant from the three vertices. In other words, if a circle is drawn with the circumcentre as its centre and radius OA or OB or OC, it would pass through all the vertices of the triangle.

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3. Angle Bisector and Incentre – An angle bisector is a line that divides the angle into two equal parts.

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In the given figure, if AF is the angle bisector, then,

\[\frac{AB}{AC}\] = \[\frac{BF}{FC}\]

and AB х AC - BF х FC = AF²

The point of intersection of the three angle bisectors of a triangle is called incentre. In the figure, G is the incentre of triangle ABC.

4. Altitude and Orthocentre – The perpendicular drawn from the vertices of the opposite sides are called altitude. The point of intersection of three altitudes is called orthocentre.

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\[\angle BOC\] = 180⁰ - \[\angle A\] 

\[\angle AOB\] = 180⁰ - \[\angle C\] 

\[\angle AOC\] = 180⁰ - \[\angle B\] 

Angle of Elevation

Let O and P be two points; such that P is at a higher level than O. Suppose O is the position of the observer and P is the position of the object. Then, OP is called the line of observation or line of sight. \[\angle POM\] = θ denotes the angle of elevation of P as observed from O.

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Angle of Elevation

Let O and P be two points; such that P is at a lower level than O. Suppose O is the position of the observer and P is the position of the object. Then, OP is called the line of observation or line of sight. \[\angle MOP\] = θ denotes the angle of depression of P as observed from O.

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Solved Examples:

Example 1:

From a cliff 150 m above the sea level, the angle of depression of a ship is 19⁰30’. What is the distance from the ship to a point on the shore directly below the observer?

Solution:

Let OB be the cliff of height 150 m, and A be the position of the ship such that the angle of depression of the ship is

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In triangle OAB,

tan 19⁰30’ = \[\frac{OB}{OA}\]

OA = \[\frac{OB}{tan 19^{0}30^{’}}\]

OA = \[\frac{150}{0.3541}\]

OA = 423.60 m

Example 2:

What is the length of diameter of the circumcircle of a triangle with sides 5 cm, 6 cm and 7 cm?

Solution:

Area of triangle is calculated as:

\[\sqrt{s(s-a)(s-b)(s-c)}\]

Where,

s = \[\frac{a+b+c}{2}\] = \[\frac{5+6+7}{2}\] = 9

Therefore,

Area = Δ = \[\sqrt{9(9-5)(9-6)(9-7)}\] = \[\sqrt{216}\] = 6\[\sqrt{6}\]

Now,

Radius of a circumcircle is calculated as:

R = \[\frac{abc}{4\Delta }\] = \[\frac{5.6.7}{4.6\sqrt{6}}\] = \[\frac{35}{4\sqrt{6}}\]

Diameter = 2R = 2.\[\frac{35}{4\sqrt{6}}\] = \[\frac{35}{2\sqrt{6}}\]

Did You Know

• The median drawn from a point to the opposite side of a triangle is also the perpendicular bisector of the side.

• In an isosceles triangle, the centroid, the orthocentre, the circumcentre and the incentre lie on the altitude.

• The median, angle bisector, altitude and perpendicular bisector of sides are all represented by same straight lines if the triangle is equilateral.

• In an equilateral triangle,

Area is given by \[\frac{\sqrt{3}}{4}\]a\[^{2}\] 

Altitude is given by \[\frac{\sqrt{3}}{2}\]a  

Circum-radius is given by \[\frac{a}{\sqrt{3}}\]

In-radius is given by \[\frac{a}{2\sqrt{3}}\] 

Where a is length of side.

• A circle passing through all the vertices of a triangle is called the circumcircle. The radius of the circumcircle is termed as the circumradius of the triangle.

• A circle which can be inscribed within the triangle so that it touches all the three sides is called incircle of the triangle. The radius of the incircle is termed as the inradius of the triangle. 

• Determinants can be used to calculate area of a triangle if vertices are given.