Question

If the sum of the legs of a right triangle is 49 inches and the hypotenuse is 41 inches, find the length of the other two sides

Answer 1

The lengths of the other two sides of the right triangle are 25 inches and 24 inches, or 24 inches and 25 inches, depending on the configuration of the sides.

Step-by-step explanation:

The given problem involves a right triangle, where the sum of the two legs is 49 inches and the hypotenuse is 41 inches. We can use the Pythagorean Theorem to find the lengths of the other two sides. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

Let's assume one leg is 'a' inches and the other leg is 'b' inches.

According to the Pythagorean Theorem, we have the equation: a^2 + b^2 = c^2, where 'c' is the hypotenuse.

Substituting the given values, we have: a^2 + b^2 = 41^2. Since the sum of the legs is 49 inches, we can express one leg in terms of the other: a = 49 - b.

Now, substitute this value into the equation and solve for 'b': (49 - b)^2 + b^2 = 41^2. Expanding and simplifying, we get: 2400 - 98b + 2b^2 = 0. Solving this quadratic equation, we find two possible solutions for 'b': b = 24 or b = 25.

Substituting these values back into a = 49 - b, we can find the corresponding values for 'a'. Therefore, the possible lengths of the other two sides of the right triangle are: a = 25 and b = 24, or a = 24 and b = 25.

Answer 2

The length of one leg is x inches.
The sum of the legs is 49 inches, so the length of the other leg is 49-x inches.
The length of the hypotenuse is 41 inches.

Use the Pythagorean theorem:

`(\hbox{one leg})^2 + (\hbox{the other leg})^2=(\hbox{hypotenuse})^2 \\ x^2+(49-x)^2=41^2 \\ x^2+2401-98x+x^2=1681 \\ 2x^2-98x+2401=1681 \ \ \ |-1681 \\ 2x^2-98x+720=0 \\ \\ a=2 \\ b=-98 \\ c=720 \\ b^2-4ac=(-98)^2-4 * 2 * 720=9604-5760=3844 \\ \\ x=(-b \pm √(b^2-4ac))/(2a)=(-(-98) \pm √(3844))/(2 * 2)=(98 \pm 62)/(4) \\ x=(98-62)/(4) \ \lor \ x=(98+62)/(4) \\ x=(36)/(4) \ \lor \ x=(160)/(4) \\ x=9 \ \lor \ x=40`


`49-x=49-9 \ \lor \ 49-x=49-40 \\ 49-x=40 \ \lor \ 49-x=9`

The lengths of the legs are 9 inches and 40 inches.