Properties of Triangle OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-7. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-7. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.
Properties of Triangle OP Malhotra S.Chand ISC Class-11 Maths Solutions
| Class: | 11th |
| Subject: | Mathematics |
| Chapter : | Ch-7 Properties of Triangle of Section -A |
| Board | ISC |
| Writer | OP Malhotra |
| Publications | S.Chand Publications 2020-21 |
-: Select Topics :-
Properties of Triangle OP Malhotra S.Chand ISC Class-11 Maths Solutions
Properties of Triangle
Let us take a triangle in which we represent ∠BCA = C, ∠ABC = B, ∠BAC = A and the side opposite to CA, AB and BC be represented by b, c and a respectively. We write the perimeter as 2s and the semi-perimeter s be represented by S = (a+b+c)/2. The area of the triangle is presented by Δ. R represents the radius of the circum-circle and r represents the radius of the in-circle.
We know,
∠BCA + ∠ABC + ∠BAC = Π or C + B + A = Π
Sine Rule:
It states a/sin A = b/sin B = c/sin C = 2R
or we can write this as the sides of the triangle are proportional to the sine of the opposite angles respectively.
From this, we can also write as a = 2R × sin A, b = 2R × sin B, c = 2R × sin C
We can also write this as, a cosA + b cosB + c cosC = 4R sinA sinB sinC
Types of Triangle
| Based on the Sides | Based on the Angles |
| Scalene Triangle | Acute angled Triangle |
| Isosceles Triangle | Right angle Triangle |
| Equilateral Triangle | Obtuse-angled Triangle |
- Scalene Triangle: All the sides and angles are unequal.
- Isosceles Triangle: It has two equal sides. Also, the angles opposite these equal sides are equal.
- Equilateral Triangle: All the sides are equal and all the three angles equal to 60°.
- Acute Angled Triangle: A triangle having all its angles less than 90°.
- Right Angled Triangle: A triangle having one of the three angles exactly 90°.
- Obtuse Angled Triangle: A triangle having one of the three angles more than 90°.
Triangle Formula
- Area of triangle is the region occupied by a triangle in a two-dimensional plane. The dimension of the area is square units. The formula for area is given by;
Area = 1/2 x Base x Height
- The perimeter of a triangle is the length of the outer boundary of a triangle. To find the perimeter of a triangle we need to add the length of the sides of the triangle.
P = a + b + c
- Semi-perimeter of a triangle is half of the perimeter of the triangle. It is represented by s.
s = (a + b + c)/2
- where a, b and c are the sides of the triangle.
- By Heron’s formula, the area of the triangle is given by:
A = √[s(s – a)(s – b)(s – c)]
where ‘s’ is the semi-perimeter of the triangle.
- By the Pythagorean theorem, the hypotenuse of a right-angled triangle can be calculated by the formula:
Hypotenuse2 = Base2 + Perpendicular2
Exe-7
Properties of Triangle OP Malhotra S.Chand ISC Class-11 Maths Solutions
Page 7-8 to 7-10
Question 1:
In ΔABC ,
(i) If a= 2 , b=3 , c=4 , prove that cos A = 7/8.
(ii) if ……………………… is 30 degree.
(iii) if a=9 , ……………………….. cos B.
(iv) The sines …………………………….. 12 : 9 : 2.
(v) if the two …………………………….. the triangle.
(vi) if in a ………………………………… find its area.
(vii) In a triangle…………………………… find the ∠ ABD.
Question 2:
………………………….
……………………………
……………………………..
Question 7:
a cos (A+ B + C )…………………….. =0
Question 8:
…………………………..
………………………….
…………………………..
Question 16:
a³ sin (B-C)……………………………… =0
Question 17:
………………………….
………………………….
………………………….
Question 21:
If sin 2A + sin 2 B…………………………….. = 90 degree
Question 22:
………………………….
…………………………….
Question 26:
The angle ………………………….. (√5-1)
…………………………
Question 28:
Two sides and included …………………………… 3√2.
Question 29:
In a ΔABC, if B = 3 C , prove that,
(i) cos C = …………….
(ii) sin A/2 ………………………..
-: End of Properties of Triangle Solution :-
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