How can one find the area of a triangle ABC whose vertices are A(4,4), B(0,0), and C(6,2)?
To find the area of a triangle ABC whose vertices are A(4,4), B(0,0), and C(6,2), we’ll first utilize the distance formula, which is based on the Pythagorean Theorem, to calculate the lengths (a, b, and c) of the three sides of the triangle and then use this information in Heron’s formula to finally find the area.Now, finding the length of each of the 3 sides of triangle ABC:Side BC:Side BC joins vertices B(0,0) and C(6,2).Since side BC is opposite angle A, then we’ll designate the length of side BC as “a”.Using the distance formula as follows:d = √[(x2 – x1)² + (y2 – y1)²]a = √[(6 – 0)² + (2 – 0)²]Note: The coordinates of the vertex of angle C could have been designated as (x1, y1) and the coordinates of the vertex of angle B could have been designated as (x2, y2). The important thing is is that once you make a choice, BE CONSISTENT; otherwise, you’ll could very well get an erroneous final answer for the area of triangle ABC!a = √[(6)² + (2)²]a = √(36 + 4)a = √40a = √[(4)(10)]a = 2√10In a very similar fashion, we’ll find the lengths of each of the other two sides AC and AB as follows:Side AC:Side AC joins vertices A(4,4) and C(6,2).Since side AC is opposite angle B, then we’ll designate the length of side AC as “b”.Using the distance formula as follows:d = √[(x2 – x1)² + (y² – y1)²]b = √[(6 – 4)² + (2 – 4)²]b = √[(2)² + ( – 2)²]b = √[(2)(4)]b = 2√2 Side AB:Side AB joins vertices A(4,4) and B(0,0).Since side AB is opposite angle C, then we’ll designate the length of side AB as “c”.Using the distance formula as follows:d = √[(x2 – x1)² + (y2 – y1)²]c = √[(4 – 0)² + (4 – 0)²]c = √[(4)² + (4)²]c = √[(16)(2)]c= 4√2Now, using Heron’s Formula to find the area of triangle ABC:Area A = √[s(s – a)(s – b)(s – c)], where s is one-half the perimeter p of triangle ABC, i.e., s = (½)(a + b+ c) = (½)(2√10 + 2√2 + 4√2) = (½)(2)(√10 + √2 + 2√2)s = √10 + √2 + 2√2s = √10 + 3√2 Now, substituting into Heron’s Formula, we have:A = √[s(s – a)(s – b)(s – c)]= {(10 + 3√2)[(√10 + 3√2) – 2√10][(√10 + 3√2) – 2√2][(√10 + 3√2) – 4√2]}^(½) = {[(3√2 + √10)(3√2 – √10)][(√10 + √2)(√10 – √2)]}^(½)= [(18 – 10)(10 – 2)]^(½)= √[(8)(8)]= √(64)Area A = 8 square units for triangle ABC.
If in triangle abc a=5 b=6 c=8 then cos a=?
Use the law of cosines. a^2= b^2+c^2-2(b)(c)cosA (5)^2= (6)^2+(8)^2-2(6)(8)cosA 25= 36+64-96cosA 25=100-96cosA -75=-96cosA cosA= 75/96 A= .674131 radians or 38.6248 degrees
If A:B=1:2, B:C=3:4, C:D=6:9, D:E=12:16, then A:B:C:D:E is equal to?
There is a simple way of doing these kind of problems by multiplying both sides with multiples of a number make a paticular variable common in both the ratioA:B = 1:2 , B:C = 3:4Now multiply A:B with 3 and multiply B:C with 2Then A:B = 3:6 and B:C = 6:8Now combining both A:B:C = 3:6:8Again C:D = 6:9Now multiplying A:B:C with 3 and C:D with 4Then A:B:C = 9:18:24and C:D = 24:36Now combining the above twoA:B:C:D = 9:18:24:36Again D:E = 12:16Now by multiplying A:B:C:D by 1 and D:E by 3 we get,A:B:C:D = 9:18:24:36and D:E = 36:48So again combining the above twoA:B:C:D:E = 9:18:24:36:48Since the above ratio is a multiple of 3 so after division:-A:B:C:D:E = 3:6:8:12:16
How many miles of carbon atoms are there in 4 mol of dimethylsulfoxide? A.2 B.6 C.8 D.4?
dimethyl sulfoxide = C2H6OS = 78.1334 g/mol (4 mol DMSO) x (2 mol C / 1 mol DMSO) = 8 mol C
If a+b=6, and a-b=2, what is the value of a, and b?
When I was a child, I was also stuck in that kind of equations. I did not know how to solve them. But now I am in B.Sc 2nd Year and this kind of equations I solve daily (more than 100). So here is your answer Step by Step. Let's start with your equations . - a + b = 6 --------- (1) a - b = 2 ----------(2)First we add both equations.adding equations (1) and (2),We get -(a+b) + (a-b) = 6+2 a + b + a - b = 8 This 'b' and '-b' becomes zero after addition,Then we get - 2a = 8 a = 8/2 a = 4 (We get the value of a)Now we get the value of a. Let's move on 'b'First we put value of 'a' in equation (1)Put value of 'a' in equation (1) 4 + b = 6 b = 6 - 4 b = 2 (That's the value of b)That's it ...........