The length of the missing side is 16.1 cm to 1 decimal place.
To find the length of the missing side in the triangle, we can use the law of cosines.
Law of cosines:
c² = a² + b² - 2ab * cos(C)
where:
c is the length of the hypotenuse of the triangle
a and b are the lengths of the other two sides of the triangle
C is the angle between sides a and b
Using the law of cosines:
In the given triangle, we are given the following:
c = 10 cm (hypotenuse)
a = 13 cm (one of the legs)
b = x (the missing leg)
C = 40° (the angle between sides a and b)
Substituting these values into the law of cosines, we get:
10² = 13² + x² - 2 * 13 * x * cos(40°)
100 = 169 + x² - 11.7 * x
-69 = x² - 11.7 * x
Solving the quadratic equation:
We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where:
a = 1
b = -11.7
c = -69
Substituting these values into the quadratic formula, we get:
x = (11.7 ± √(11.7² - 4 * 1 * -69)) / 2 * 1
x = (11.7 ± √(139.29 + 276)) / 2
x = (11.7 ± √415.29) / 2
x = (11.7 ± 20.4) / 2
Therefore, the two possible solutions for x are:
x = 16.1 cm or x = -8.7 cm
Since the length of a side cannot be negative, the only valid solution is x = 16.1 cm.