triangle ABC, AB = x cm, BC = (x-4)cm, AC =10 cm and angle BAC = 60° calclulate the value of x
Triangle ABC
AB = x
BC = x - 4
AC = 10
angle BAC = 60 deg
Using Law of Cosines
a^2 = b^2 + c^2 - 2bc cos A
(x -4)^2 = 10^2 + x^2 - 2(10)(x)cos 60
x^2 - 8x + 16 = 100 + x^2 - 20x (0.5)
x^2 - 8x + 16 = 100 + x^2 - 10x
2x = 84
x = 42
check out law of cosines
To find the value of x, we will use the law of cosines. The law of cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where a, b, and c are the side lengths of the triangle, and C is the angle opposite side c.
In this case, side c is AC (10 cm), side a is AB (x cm), and side b is BC (x-4 cm). Angle C is angle BAC (60°).
Plugging in the values into the formula:
(10 cm)^2 = (x cm)^2 + (x-4 cm)^2 - 2(x cm)(x-4 cm) * cos(60°)
Simplifying and solving for x:
100 cm^2 = x^2 + (x-4)^2 - 2x(x-4)(1/2)
100 cm^2 = x^2 + (x-4)^2 - x(x-4)
100 cm^2 = x^2 + x^2 - 8x + 16 - x^2 + 4x
100 cm^2 = x^2 - 4x + 16
Rearranging the equation:
x^2 - 4x + 16 - 100 cm^2 = 0
x^2 - 4x - 84 cm^2 = 0
Now we can solve this quadratic equation for x. By factoring, completing the square, or using the quadratic formula:
(x - 14)(x + 6) = 0
Setting each factor equal to zero:
x - 14 = 0 --> x = 14
x + 6 = 0 --> x = -6
Since the length of a side cannot be negative, the value of x is 14 cm.
Therefore, the value of x in triangle ABC is 14 cm.
To find the value of x, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c and angle C opposite side c, we have the following equation:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we have triangle ABC with side AB = x cm, BC = (x-4) cm, and AC = 10 cm. We are trying to find the value of x.
Applying the law of cosines, we can write the equation as follows:
(10 cm)^2 = (x cm)^2 + (x-4 cm)^2 - 2(x cm)(x-4 cm) * cos(60°)
Simplifying the equation, we have:
100 cm^2 = x^2 + (x-4)^2 - 2x(x-4) * cos(60°)
Expanding and simplifying further:
100 cm^2 = x^2 + x^2 - 8x + 16 - 2x^2 + 8x * cos(60°)
100 cm^2 = -x^2 + 8x + 16 + 8x * cos(60°)
100 cm^2 = -x^2 + 8x + 16 + 8x * (0.5)
100 cm^2 = -x^2 + 8x + 16 + 4x
100 cm^2 = -x^2 + 12x + 16
Rearranging the equation:
x^2 - 12x + 84 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -12, and c = 84. Plugging in these values into the quadratic formula:
x = (-(-12) ± √((-12)^2 - 4(1)(84))) / (2(1))
x = (12 ± √(144 - 336)) / 2
x = (12 ± √(-192)) / 2
Since the value under the square root is negative, the equation has no real solutions. Therefore, there is no value of x that satisfies the given conditions.