the hypotenuse of a triangle is square root of 40 in. long. the length of one leg is 2 in. less than four times the other leg. find the lengths of the legs.

Answer 1

Although you do not explicitly say that the triangle is a right triangle, the non-right triangles do not have a hypotenuse. Another point is that you must mean the square root of 40 square-inches.

Using the Pythagorean Formula, c² = a² + b², we have this:

c = √(40)

a = 4b -2

Noting that c² = 40, we can plug these values into the Pythagorean Formula to yield this equation:

40 = (4b - 2)² + b²

= (16b² - 16b + 4) + b²

= 17b² - 16b + 4

Subtracting 40 from both sides gives

0 = 17b² - 16b - 36

Using the quadratic formula, we get

b = [ 16 ± sqrt( 16² - 4(17)(-36) ) ] / 2(17)

b = 2 or -1.06

Since the legs of a triangle can have only positive lengths, we can eliminate the negative value, leaving b = 2.

a = 4b - 2 = 4(2) - 2 = 6

Answer: 2 and 6.

Check:

2² + 6² = 4 + 36 = 40.