Question

using pythagoreans theorem and exact answers, determine the value of x in each triangle. 10) 11) 12) 13) 14) 15) 16) what do the triangles in problems 10 - 15 all have in common? what patterns do you notice?

Explanation:

Step1: Recall 30 - 60 - 90 triangle ratios

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is $a$, the side opposite the 60° angle is $a\sqrt{3}$, and the hypotenuse is $2a$.

Step2: Solve for $x$ in each triangle

For problem 10:

The side opposite the 30° angle is 18. The hypotenuse $x = 2\times18=36$.

For problem 11:

The side opposite the 30° angle is $9\sqrt{3}$. The hypotenuse $x = 2\times9\sqrt{3}=18\sqrt{3}$.

For problem 12:

The side opposite the 30° angle is 10. The hypotenuse $x = 2\times10 = 20$.

For problem 13:

The side opposite the 30° angle is 7. The hypotenuse $x=14$.

For problem 14:

The side opposite the 30° angle is 7. The hypotenuse $x = 14$.

For problem 15:

The side opposite the 30° angle is $9\sqrt{3}$. The hypotenuse $x=18$.

Step3: Analyze common - features

All the triangles in problems 10 - 15 are right - triangles with angles 30°, 60°, and 90°. The ratio of the sides opposite the 30°, 60°, and 90° angles follow the pattern $1:\sqrt{3}:2$.

Answer:

1. $x = 36$
2. $x=18\sqrt{3}$
3. $x = 20$
4. $x = 14$
5. $x = 14$
6. $x = 18$
7. They are all 30 - 60 - 90 right - triangles and the side - length ratios follow the pattern $1:\sqrt{3}:2$.