In a certain right triangle, the lengths of the legs have a sum of 50 and a product of 282. What is the length of the hypothenuse?
A. 44
B. 42
C. 40
D. 25squareroot2
If the legs are a and b, and the hypotenuse is c, then
a+b=50
ab=282
c^2 = a^2+b^2 = (a+b)^2 - 2ab
= 2500-564
= 1936
so, c = √1936 = 44
1 2 3 6 47 94 141 282
To solve this problem, let's use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Let's assume the lengths of the legs are "a" and "b", and the length of the hypotenuse is "c".
According to the problem, the lengths of the legs have a sum of 50, so we can write the equation: a + b = 50.
We are also given that the product of the lengths of the legs is 282, so we can write another equation: ab = 282.
Now, let's solve these equations step-by-step.
Step 1: Solve the first equation for one of the variables.
We can solve the first equation for "a" in terms of "b" by subtracting "b" from both sides:
a = 50 - b.
Step 2: Substitute the value of "a" into the second equation.
Substituting 50 - b for "a" in the second equation, we get:
(50 - b)b = 282.
Step 3: Simplify and solve the quadratic equation.
Expanding the equation, we have:
50b - b^2 = 282.
Rearranging the equation, we get:
b^2 - 50b + 282 = 0.
Since this is a quadratic equation, we can solve it using factoring, completing the square, or the quadratic formula. In this case, to find the lengths of the legs, we need to use the quadratic formula. However, to find the length of the hypotenuse, we can simply use the quadratic equation in step 4.
Step 4: Use the quadratic equation to find the length of the hypotenuse.
Applying the quadratic formula, we have:
b = (-(-50) ± √((-50)^2 - 4(1)(282))) / (2(1)).
Simplifying further, we have:
b = (50 ± √(2500 - 1128))/2.
b = (50 ± √1372)/2.
b = (50 ± 37)/2.
Taking both values of "b", we get two possible values for the lengths of the legs: b = 87/2 or b = 13/2.
Since the length of a leg cannot be negative, we discard the solution b = 87/2.
Now, substituting b = 13/2 into the equation a + b = 50, we can solve for "a" as follows:
a + 13/2 = 50.
a = 50 - 13/2.
a = (100 - 13)/2.
a = 87/2.
Therefore, the lengths of the legs are a = 87/2 and b = 13/2.
Finally, let's find the length of the hypotenuse using the Pythagorean Theorem: c^2 = a^2 + b^2.
Substituting the values, we get:
c^2 = (87/2)^2 + (13/2)^2.
Simplifying, we have:
c^2 = 7569/4 + 169/4.
c^2 = 7738/4.
c^2 = 1934.
So, the square of the length of the hypotenuse is 1934. To find the length of the hypotenuse, we take the square root of 1934.
c = √1934.
Hence, the length of the hypotenuse is approximately 44.03.
Therefore, the correct answer is A. 44.
To find the length of the hypotenuse in a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
Let's assume the lengths of the legs are a and b, and the length of the hypotenuse is c. According to the problem, we know that a + b = 50 and ab = 282.
To find the length of the hypotenuse, we can solve the given equations simultaneously. Let's express one variable in terms of the other and substitute it back into the other equation.
From the equation a + b = 50, we can express b as b = 50 - a. Substituting this value into the equation ab = 282, we have a(50 - a) = 282.
Expanding the equation, we get 50a - a^2 = 282. Rearranging, we have a^2 - 50a + 282 = 0.
Now let's solve this quadratic equation for a. We can use factoring, completing the square, or the quadratic formula. Factoring doesn't seem to be an obvious option, so let's use the quadratic formula.
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by x = (-b ± √(b^2 - 4ac)) / (2a).
In our equation a^2 - 50a + 282 = 0, a = 1, b = -50, and c = 282.
Plugging these values into the quadratic formula, we get:
a = (-(-50) ± √((-50)^2 - 4(1)(282))) / (2(1))
= (50 ± √(2500 - 1128)) / 2
= (50 ± √1372) / 2
Simplifying further:
a = (50 ± √(2^2 * 343)) / 2
= (50 ± 2√343) / 2
= 25 ± √343
So we have two possible values for a: 25 + √343 and 25 - √343.
Now let's calculate the possible lengths of the hypotenuse.
If a = 25 + √343, then b = 50 - (25 + √343) = 25 - √343.
Using the Pythagorean theorem, we can find the length of the hypotenuse:
c^2 = (25 + √343)^2 + (25 - √343)^2
= (25 + √343)(25 + √343) + (25 - √343)(25 - √343)
= 25^2 + 2(25)(√343) + (√343)^2 + 25^2 - 2(25)(√343) + (√343)^2
= 2(25^2) + 2(√343)^2
= 2(625) + 2(343)
= 1250 + 686
= 1936
Taking the square root of both sides, we find:
c = √1936
= 44
Therefore, the length of the hypotenuse is 44.
Hence, the correct answer is option A.