Find the length of the third side of a triangle if the area of the triangle is 18 and two of its sides have lengths of 5 and 10.
Let the angle between the 2 given sides by Ø
then
(1/2)(5)(10)sinØ = 18
sinØ = 18/25
by Pythagoras: cosØ = √301/25
let the third side be x
by the Cosine Law
x^2 = 5^2 + 10^2 - 2(5)(10)cosØ
= 125 - 100(√301/25)
= 55.60259...
x = appr 7.46
To find the length of the third side of a triangle, we can use the Heron's formula.
Heron's formula states that the area (A) of a triangle with sides a, b, and c is given by:
A = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle and is calculated as:
s = (a + b + c) / 2
Let's use this formula to solve for the length of the third side:
Given: a = 5, b = 10, A = 18
We first need to calculate the semi-perimeter (s):
s = (a + b + c) / 2
s = (5 + 10 + c) / 2
s = (15 + c) / 2
Next, we substitute the given values into the area formula and solve for c:
A = √(s(s-a)(s-b)(s-c))
18 = √((15 + c) / 2)((15 + c) / 2 - 5)((15 + c) / 2 - 10)(c)
Squaring both sides of the equation, we get:
18^2 = ((15 + c) / 2)((15 + c) / 2 - 5)((15 + c) / 2 - 10)(c)
324 = ((15 + c) / 2)(c - 5)((15 + c) / 2 - 10)(c)
Expanding and simplifying:
324 = (c + 15)(c - 5)(c - 20)(c)
We can solve this equation using numerical methods such as graphing or approximation techniques. However, the solution is complex and cannot be conveniently done by hand. Therefore, the exact length of the third side (c) cannot be determined with the given information.
To find the length of the third side of a triangle, we can use the formula known as the "Heron's formula." This formula relates the area of a triangle to the lengths of its sides.
Heron's formula states:
Area = √(s * (s - a) * (s - b) * (s - c))
where
s = (a + b + c) / 2
a, b, and c are the lengths of the sides.
In this case, we are given that the area of the triangle is 18 and that two of its sides have lengths of 5 and 10, let's call them a and b. Therefore:
Area = 18
a = 5
b = 10
Let's substitute these values into Heron's formula:
18 = √(s * (s - 5) * (s - 10) * (s - c))
Now, we need to solve for c, which is the length of the third side. To do this, we will need to simplify the equation:
18^2 = s * (s - 5) * (s - 10) * (s - c)
324 = (s * (s - 5) * (s - 10) * (s - c))
Now, we have a quadratic equation, and we need to find the value of c that will satisfy this equation. We can expand and simplify the equation:
324 = (s^2 - 5s)(s^2 - 10s)(s - c)
Using the distributive property:
324 = (s^2 - 5s)(s^3 - 10s^2 - cs + 10c)
Expanding further:
324 = s^5 - 10s^4 - cs^3 + 10cs^2 - 5s^4 + 50s^3 + 5cs^2 - 50cs + 10cs - 100c
Combining like terms:
324 = s^5 - 15s^4 + 55s^3 + 15cs^2 - 50cs - 100c
Now, we can simplify the equation by dividing both sides by -1:
-324 = -s^5 + 15s^4 - 55s^3 - 15cs^2 + 50cs + 100c
Now, let's solve this equation using numerical methods or calculators like a graphing calculator or an equation solver.
After solving the equation, we will have the value of s and c. The value of c will be the length of the third side of the triangle.