TRIANGLES
Properties of Triangles:
1. A triangle has three sides and three interior angles
2. Sum of any two sides of a triangle is always greater than the third side.
i.e. for a triangle ABC, we have
AB + BC > AC
AB + AC > BC
AC + BC > AB
3. Sum of all angles of a triangle is 180.
i.e For a triangle ABC
∠A + ∠B + ∠C = 180.
4. The exterior angle of a triangle is equal to the sum of the vertically
opposite interior angles.
5. If an angle of a triangle is greater than 90 then the other two angles of
triangle must be acute angles.
Types of triangle
1. Scalene : Scalene triangle is a triangle that has length of its three sides
different from one other.
(i). All angles are different. None of angles are equal.
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2. Isosceles: Isosceles triangle is a triangle that has two of it's sides equal to
each other.
(i) The angles opposite to equal sides are also equal.
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3. Equilateral: Equilateral triangle has all its sides equal to each other
(i) All angles are equal and are of 60 each.
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4. Right Angled: One of the angle of the triangle
is equal to 90.
(i). The side opposite to the right angle is greatest side in the
triangle.
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Terms related to Triangles:
1. Median and centroid : Median is a line that joins a vertex of a triangle to
the mid-point of the opposite side. Median divides the
triangle into two equal halves.If we draw all the medians of a
triangle then they will intersect at a point. That point is called
a centroid.
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2. Height and orthocentre: A straight line from any vertex of the triangle to
the opposite side and perpendicular to it , is called as a height
or altitude. The point where all the altitudes meet is called as
circumcentre.
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3. Perpendicular bisector and circumcentre : If we draw perpendicular
bisector of each side, then they will meet at a point known as
circumcentre of the triangle. Circumcentre because if we put
compass on the circumcentre and open the mouth to match the
distance between the circumcentre and any vertex, then we can
be able to draw a circle that contains all the vertices of the
triangle.
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Formula for calculating area of triangles:
1. Using base and the height:
Area of triangle = ½ x b x h
where, b is the length of base or the side of the triangle where the
altitude from the opposite vertex was drawn
h is the height of the altitude from the vertex to the base.
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2. Using Hero's formula
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Area of triangle = √s(s-a)(s-b)(s-c)
where, s = (a +b + c)/2
and a , b, c are the length of the sides of the triangle.
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3. For equilateral Triangle
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Area = (√3 x a² )
4
where, a is the side of the equilateral triangle
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4. For isosceles Triangle
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Area = b √4a² - b²
4
where, a is the length of the equal sides
and b is the length of the third side.
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Solved Questions:
1. The perimeter of an equilateral triangle is 72√3 cm. Find its height.
(1) 63 m
(2) 24 m
(3) 18 m
(4) 36 m
Solution: Let Perimeter of equilateral triangle = 3 a = 72 √3
=> a = 72√3/ 3 => a = 24√3
we also know,
Height of equilateral triangle = a (√3 / 2)
= 24 √3(√3/2)
= (24 x 3) /2
= 12 x 3
= 36
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2. In the triangle below angle ∠PQR and angle ∠QRP have the measure of 4m° , ∠QSP have the measure of m° and ∠RPS has a measure of 45°, then what is the measure of ∠QPS?
(1) 45°
(2) 60°
(3) 90°
(4) 105°
(5) 120°
Solution: ∠QRP is an exterior angle to triangle PRS.
Therefore, ∠QRP = ∠RPS + ∠RSP
=>∠QRP = 45° + m
But ∠QRP = 4m (given)
=> 4m = 45° + m
=> 3m = 45°
=> m = 15°
Thus,
∠RSP = 15°
and ∠PQS = 4 x 15 = 60°
Now in triangle PQS, We know that sum of all angles of a triangle =180
Therefore, ∠QPS + ∠PQS + PSR = 180°
=> ∠QPS = 180 - 15 - 60°
=> ∠QPS = 105°
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3 : In the triangular shown below, if x = √6, then what is the area of the triangle?
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(1) √ 3/8 (2) 3/8
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(3) 3√ 3/8 (4) 3/4
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(5) 3√ 3/4
Solution:
Given,
x = √6
To find,
Area of the given triangle
Since the triangle is a right angled triangle, then using the Pythagoras theorem, we have
i.e. (x)² = (x √3/2)² + y²
where 'y' is the base of the triangle in the figure.
=> (√6)² = {(√6 x √3)/2}² + y²
=> (√6)² = {(√18)/2}² + y²
=> 6 = {(3√2)/2}² + y²
=> 6 = (3/√2)² + y²
=> 6 = 9/2 + y²
=> y² = 6 - 4.5
=> y² = 1.5 = 3/2
=> y =√ (3/2)
Area of triangle = ½ x b x h
Putting in the values
Area of triangle = ½ x √ (3/2) x (3/√ 2)
= ½ x (3√ 3/2)
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= 3√ 3/4
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