For a triangle ABC it is given that cosA + cosB + cosC = 3/2. Prove that the triangle is equilateral
rithik , 4 Years ago
Grade 11
10 grade maths> For a triangle ABC it is given that cosA ...
1 Answersrithik
Last Activity: 4 Years ago
Let a, b, c are the sides of a ∆ ABC. Given, cos A + cos B + cos C = 3/2 ⇒ (b2 + c2 - a2)/2bc + (a2 + c2 - b2)/2ac + (a2 + b2 - c2)/2ab = 3/2 ⇒ ab2 + ac2 – a3 + ba2 + bc2 – b3 + ca2 + cb2 – c3 = 3abc ⇒ a(b – c)2 + b(c – a)2 + c(a – b)2 = (a + b + c)/2[(a - b)2 + (b - c)2 + (c - a)2] ⇒ (a + b – c) (a – b)2 + (b + c – a) (b – c)2 + (c + a – b) (c – a)2 = 0 (as we know, a+b–c > 0, b+c–a > 0, c+a–b > 0) ∴ Each term on the left of equation has positive coefficient multiplied by perfect square, each term must be separately zero. ⇒ a = b = c ∴ Triangle is an equilateral
Provide a better Answer & Earn Cool Goodies
Enter text here...
LIVE ONLINE CLASSES
Prepraring for the competition made easy just by live online class.
Full Live Access
Study Material
Live Doubts Solving
Daily Class Assignments
Ask a Doubt
Get your questions answered by the expert for free
Enter text here...
Other Related Questions on 10 grade maths
What are the advantages and disadvantages of mean, median and mode?
10 grade maths
1 Answer Available
Last Activity: 1 Year ago(s)
Find the sum of the first 100 natural numbers.
10 grade maths
1 Answer Available
Last Activity: 2 Year ago(s)
AOBC is a rectangle whose three vertices are A(0,-3), O(0,0) and B (4,0). The length of its diagonal is__
10 grade maths
1 Answer Available
Last Activity: 2 Year ago(s)
For an AP, it is given that first term (a)= 5 and Common Difference (d) = 3 and nth term = 50. Find n and sum of first n terms of AP
10 grade maths
1 Answer Available
Last Activity: 2 Year ago(s)
The value(s) of k for which the quadratic equation 2x^2 + kx + 2 = 0 has equal roots, is (a) 4 (b) pm 4 (c) -4 (d) 0
10 grade maths
1 Answer Available
Last Activity: 2 Year ago(s)