Question

what is the length of x in the diagram? 45° 5 30° x

Explanation:

Step1: Find the length of the common - side

In the left - hand 45 - 45 - 90 triangle, if the base is 5, and for a 45 - 45 - 90 triangle with legs \(a\) and hypotenuse \(c\), the ratio of the sides is \(a:a:c = 1:1:\sqrt{2}\). Here, the length of the common side (the non - hypotenuse side of the 45 - 45 - 90 triangle) is 5.

Step2: Use trigonometry in the right - hand 30 - 60 - 90 triangle

In the right - hand 30 - 60 - 90 triangle, the side opposite the 30 - degree angle is 5. For a 30 - 60 - 90 triangle with sides \(a\) (opposite 30 degrees), \(b\) (opposite 60 degrees), and \(c\) (hypotenuse), the ratio of the sides is \(a:b:c=1:\sqrt{3}:2\). If the side opposite the 30 - degree angle is \(a = 5\), and the hypotenuse is \(x\), and the relationship between the side opposite 30 degrees and the hypotenuse is \(a=\frac{1}{2}c\). So, \(x = 10\).

Answer:

10