A right triangle has a hypotenuse of 5 ft. If one of the legs is 1 ft. longer than the other,what are the lengths of the two legs?

Question

A squared + b squared= c squared

Answer 1

a =4 ft

b = 3 ft

A squared + b squared= c squared

Per the information given, we can let a = b +1 this gets the Pythagorean Theorem in terms of one variable, b. We can just substitute into the equation and solve for b.

c = hypotenuse = 5

a = b + 1

b = b

Make the substitution below

a² + b² = c²

(b +1)² + b² = c²

PEMDAS

b² + 2b + 1+ b² = 25

Combine like terms

2b² + 2b + 1 = 25

Subtract 25 from both sides of the equation, this forms the quadratic

2b² + 2b - 25+ 1 = 0

Combine like terms

2b² + 2b - 24 = 0

Factor the quadratic

(2b + 8)(b - 3) = 0

b= -4, b = 3

b cannot be negative so

Leg b of the right triangle is 3 feet

b = 3

a = b + 1= 3 + 1 = 4

a = 4

b = 3

Checking

a² + b² = c²

4² + 3² = 5²

16 + 9 = 25

I hope you find this useful if you have any questions send me a message.

Answer 2

So the pythagorean theorem for a right triangle (that is, a triangle with one interior angle of 90 deg.) is

a² + b² = c²

Where a and b are the lengths of the sides adjacent to the 90 deg angle and c is the length of the hypotenuse (which is opposite to the 90 deg angle and also the longest side. These two properties are connected by the way!)

Using algebra, we can solve this problem. If c=5 and a=x, then we must set b=x+1. Now we can just plug these values into our pythagorean theorem for right triangles and solve for x.

a² + b² = c²

x² + (x+1)² = 5²

x² + x² + 2x + 1 - 25 = 0

2x² + 2x - 24 = 0

x² + x - 12 = 0

(x - 3) (x + 4) = 0

x = {-4, 3}

Neglecting the negative value, we get that one side length must be 3, and the other must be 4 under the given conditions. Also note that this is a 3-4-5 triangle, a very famous right triangle in geometry and trigonometry.

Thank you for your question and I hope this helps!

-MJM