How do you solve this trigonometry question?

Question

The length of one side of the right triangle is 10cm. What might the lengths of the other sides be? Provide 5 solutions 

Answer 1

Use the Pythagorean triples.  If the side of the given triangle is similar to a 3-4-5 triangle, then the sides make the triangle a right triangle.

 

3² + 4² = 5²    ---->   Pythagorean triple

 

 

If 10 is the hypotenuse,

 

x² + y² = 10²

 

 

Using a proportion,

 

x / 3 = y / 4 = 10 / 5

 

x / 3 = y / 4 = 2

 

 

Solving for x gives            x = 6       

 

and   

 

 

Solving for y gives             y = 8

 

 

One possibility is 6-8-10 triangle.

 

 

If 10 is the leg of a triangle,

 

x² + 10² = c²

 

3² + 4² = 5²

 

and you know that the sum of two legs is larger than the longest side.  The longest side is the hypotenuse.

 

x + 10 > c

 

 

Using a proportion,

 

x / 3 = 10 / 4 = c / 5

 

x / 3 = 5 / 2 = c / 5

 

 

Solving for x,  we use the equation

 

x / 3 = 5 / 2

 

x = (5 / 2) * 3

x = 15/2

x = 7.5

 

 

Solving for c,

 

c /5 = 5 / 2

 

c = (5/2) * 5

c = 25/2

c = 12.5

 

Your sides are 7.5 , 10, and 12.5

 

 

There are only 4 possible solutions:

 

1)   6 cm , 8 cm , 10 cm

2)   8 cm , 6 cm , 10 cm

3)  7.5 cm , 10 cm , 12.5 cm

4)  10 cm , 7.5 cm  , 12.5 cm

 

(1) and (2) both have 10 cm as the hypotenuse.  Only the legs switch orientation.

(3) and (4) both have 12.5 cm as the hypotenuse.  10 cm could be the larger leg or the shorter leg

 

 

Answer 2

There are infinite answers to this question.

 

From the Pythagorean Theorem:  a² + b² = c²

where a, b are the leg lengths and c is hypotenuse

of a right triangle.

 

10² + b² = c²

100 + b² = c²

 

You can pick any length for side b you desire

and from that find the corresponding length

of the hypotenuse