Answer:
x = 20√6 units
Explanation:
Determining the measure of the third angle in the two triangles:
Let's call the triangle with the 60° angle Triangle A and the triangle with the 45° angle Triangle B.
In any triangle, the sum of the interior angles is always 180.
Note that both triangles have a right angle, whose measure is 90°
Therefore, we know that the measure of the third angle in Triangle A is 30° as 30 + 60 + 90 = 180, which means its a 30-60-90 triangle.
Similarly, we know that the measure of the third angle in Triangle B is 45° as 45 + 45 + 90 = 180, which means its a 45-45-90 triangle.
Rules for a 30-60-90 triangle:
A 30-60-90 triangle has the following rules concerning its sides:
- The side opposite the 30° angle is the shortest side and we can call its length "x" units.
- The side opposite the 60° angle is larger than the 30° side and we can call its length "x√3" units.
- The side opposite the 90° (right) angle (the side opposite the 90° angle is formally called the hypotenuse) is the largest side and we can call its length 2x.
Determining the length of the side opposite the 90° angle:
Since the side opposite the 30° angle is 10√3, we can call it x.
Since the side opposite the 90° angle is "2x" units, we can find it by doubling 10√3:
2(10√3)
20√3
Thus, the side opposite the 90° angle is 20√3 units.
Note that this side is also a side in Triangle B.
Rules for a 45-45-90 triangle:
A 45-45-90 triangle has the following rules concerning its sides:
- The sides opposite the 45° angles are congruent and we can call their lengths "x" units.
- The side opposite the 90° (right) angle is the longest side called the hypotenuse and we can call its length "x√2" units.
We know that x for Triangle B is 20√3 units as the side opposite the 45° angle is the same side opposite the 90° angle in Triangle A.
Thus, we can multiply 20√3 by √2 to find x:
(20√3)(√2)
20√6.
Thus, x = 20√6 units.