A triangle has vertices A(2,3,7), B(0,-3,4) and C(5,2,-4)
A) determine the largest angle in the traingle
b) determine the area of of the triangle
pls tell me how to solve the problem
Now let's see if one of those angles is 90 degrees because that would make the area easy.
dot product is magnitudes *cos angle
AB = -2 i -6 j -3 k
AC = +3 i -1 j -11k
AB dot AC = -6+6 +33 = +33
we already know |AB|=c = 7, |AC|=b = 11.4
so
7*11.4* cos A = 33
cos A = .4135
A = 65.6 degrees
Now find angle B
BA = -AB so
BA = 2 i + 6 j + 3 k
BC = 5 i + 5 j - 8 k
BA dot BC = 10 + 30 - 24 = 16
7*10.7* cos B = 16
cos B = .2136
B = 77.7 degrees
Oh well not a right triangle
We could find angle C by subtraction from 180 but calculate it as a check.
CA = - AC
CA = -3 i + 1 j +11 k
CB = -5 i - 5 j + 8 k
10.7*11.4 cos C = 15-5+88 =98
cos C = .8034
C = 36.5 degrees
check
36.5 + 77.7 + 65.6 = 179.8 which is close enough to 180
Now you have three sides and the opposite angles. I will leave it to you to find the area.
a opposite angle A etc
a^2 = 5^2 + 5^2 + 8^2 = 114
a = 10.7
b^2 = 3^2 + 1^2 +11^2 = 131
b = 11.4
c^2 = 2^2 + 6^2 + 3^2 = 49
c = 7
so side b is biggest
let AB represent vector AB, and AC as vector AC
AB = (-2,-6,-3), |AB| = 7
AC = )3,-1,-11) |AC| = √131
AB•AC = |AB||AC|cos Ø
cosØ = 33/(7√131)
Ø = 65.676
area = (1/2)(AB)(AC)sinØ
= 36.5
A) To determine the largest angle in the triangle, we can use the dot product formula.
The dot product of two vectors A and B is given by the formula: A · B = |A| |B| cosθ, where θ is the angle between A and B.
Let's find the lengths of the sides first:
Side AB = √[(0 - 2)^2 + (-3 - 3)^2 + (4 - 7)^2] = √[(4 + 36 + 9)] = √49 = 7
Side BC = √[(5 - 0)^2 + (2 + 3)^2 + (-4 - 4)^2] = √[(25 + 25 + 64)] = √114 = 10.68
Side CA = √[(2 - 5)^2 + (3 - 2)^2 + (7 + 4)^2] = √[(9 + 1 + 121)] = √131 = 11.45
Now, we can find the angles using the dot product formula. Let's find the dot products of the corresponding sides:
Dot product AB · BC = (0 - 2)(5 - 0) + (-3 - 3)(2 + 3) + (4 - 7)(-4 - 4) = -10 + (-18) + 24 = -4
Dot product BC · CA = (5 - 0)(2 - 5) + (2 + 3)(3 - 2) + (-4 - 4)(7 - 4) = -15 + 5 + (-11) = -21
Dot product CA · AB = (2 - 5)(0 - 2) + (3 - 2)(-3 - 3) + (7 + 4)(4 - 7) = 6 + (-6) + (-33) = -33
To find the angles, we'll use the formula: cosθ = (dot product of sides) / (product of their lengths)
Angle A = arccos[(BC · CA) / (|BC| * |CA|)] = arccos[-21 / (10.68 * 11.45)]
Angle B = arccos[(CA · AB) / (|CA| * |AB|)] = arccos[-33 / (11.45 * 7)]
Angle C = arccos[(AB · BC) / (|AB| * |BC|)] = arccos[-4 / (7 * 10.68)]
Now, we just need to calculate the angles using a calculator or software.
B) To determine the area of the triangle, we can use Heron's formula:
s = (AB + BC + CA) / 2 = (7 + 10.68 + 11.45) / 2 = 14.06
Area = √[s * (s - AB) * (s - BC) * (s - CA)] = √[14.06 * (14.06 - 7) * (14.06 - 10.68) * (14.06 - 11.45)]
Now, let me give you the answer with a humorous twist:
A) The largest angle in the triangle is the one with the crankiest vertex - the one who woke up on the wrong side of the Cartesian plane! Seriously though, we'll need to use some trigonometry to calculate the exact value.
B) Ah, the area of the triangle! It's like the little patch of grass in a city. It may not be much, but it brings a touch of nature to the concrete jungle! Let's plug in the values and calculate it using Heron's formula.
Now, if you want the actual numerical answers, I'd be happy to assist you further!
To solve this problem, we can use the principles of geometry and basic trigonometry. Here are the steps to find the largest angle in the triangle and compute the area:
A) Determine the largest angle in the triangle:
1. Calculate the length of each side of the triangle using the distance formula.
- Side AB: √[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]
- Side BC: √[(x3 - x2)² + (y3 - y2)² + (z3 - z2)²]
- Side AC: √[(x3 - x1)² + (y3 - y1)² + (z3 - z1)²]
2. Use the law of cosines to calculate each angle.
- Angle A = arccos[((AC)² + (AB)² - (BC)²) / (2 * AC * AB)]
- Angle B = arccos[((AB)² + (BC)² - (AC)²) / (2 * AB * BC)]
- Angle C = arccos[((BC)² + (AC)² - (AB)²) / (2 * BC * AC)]
3. Compare angles A, B, and C to identify the largest angle in the triangle.
B) Determine the area of the triangle:
1. Calculate the vectors representing the triangle's sides.
- Vector AB = B - A = (x2 - x1, y2 - y1, z2 - z1)
- Vector AC = C - A = (x3 - x1, y3 - y1, z3 - z1)
2. Compute the cross product of AB and AC vectors to find the normal vector of the triangle, which has magnitude equal to the area of the triangle.
- Normal Vector = AB × AC
3. Calculate the magnitude of the normal vector using the length formula:
- Area = |Normal Vector| = √[x² + y² + z²]
Now, let's perform the calculations using the given coordinates A(2,3,7), B(0,-3,4), and C(5,2,-4).