Can you help please on the following question:
"The sum of the lengths of the legs of a right triangle is 8 feet. If the length of the hypotenuse is 6 feet, find the length of the shorter leg to the nearest tenth of a foot."
Thank you, Cassandra
We know that a^2 + b^2 = 6^2 since the hypotenuse is 6 ft and this is pythagoras equation (^2 means squared or power of 2).
We also know that 6 + a + b = 8 if we name the unknown sides of the triangle a and b.
Therefore we can isolate either one, let's take a.
a = 8 - 6 - b so a = 2-b
if we then insert 2-b for a in the pythagoras equation we get: (2-b)^2 + b^2 = 6^2
Solve for b and you'll know a since we already established that a + b = 8
EDIT: a + b = 2 not 8, sorry if I confused you
No worries. I will review your answer today and let you know if I need further help. Thank you Mr. Amg.
leg #1 --- x
leg #2 --- 8-x
x^2 + (8-x)^2 = 6^2
x^2 + 64 - 16x + x^2 = 36
2x^2 - 16x + 28= 0
x^2 - 8x + 14 = 0
x^2 - 8x + 16 = -14 + 16 --- I am completing the square
(x-4)^2 = 2
x-4 = ± √2
x = 4+√2 or x=4-√2
x = appr 5.41 or x = appr 2.59
sides are 5.41 and 2.59
or
sides are 2.59 and 5.41, (this is called a symmetric quadratic solution)
the shorter side is 2.59 ft
Of course, Cassandra! Let's solve the problem step by step.
We have a right triangle, which means one of the angles in the triangle is a right angle (measuring 90 degrees). 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. This is known as the Pythagorean theorem.
Let's assume the lengths of the two legs are a and b. According to the problem, the sum of the lengths of the legs is 8 feet, so we can set up the equation:
a + b = 8 -- Equation 1
We are also given that the length of the hypotenuse is 6 feet. Using the Pythagorean theorem, we know that:
a^2 + b^2 = c^2
Where c represents the length of the hypotenuse. In this case, c = 6. So we can set up another equation:
a^2 + b^2 = 6^2
a^2 + b^2 = 36 -- Equation 2
Now we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and b). We can solve this system of equations to find the values of a and b.
To solve it, we can use the substitution or elimination method. In this case, let's solve it using the substitution method.
From Equation 1, we can solve for a:
a = 8 - b
Now we substitute this value of a into Equation 2:
(8 - b)^2 + b^2 = 36
Expanding (8 - b)^2:
64 - 16b + b^2 + b^2 = 36
Combine like terms:
2b^2 - 16b + 28 = 0
Divide all terms by 2 to simplify the equation:
b^2 - 8b + 14 = 0
To solve this quadratic equation, we can use the quadratic formula:
b = (-(-8) ± √((-8)^2 - 4(1)(14))) / (2(1))
Simplifying:
b = (8 ± √(64 - 56)) / 2
b = (8 ± √8) / 2
b = (8 ± 2√2) / 2
b = 4 ± √2
Now we have two possible values for b: 4 + √2 and 4 - √2. Since the problem asks for the length of the shorter leg, we choose the smaller value:
b = 4 - √2
To find the length of the shorter leg (a), we substitute this value of b back into Equation 1:
a + (4 - √2) = 8
Simplifying:
a = 8 - 4 + √2
a = 4 + √2
So the length of the shorter leg (to the nearest tenth of a foot) is approximately 4.4 feet.
I hope this explanation helps you understand how to solve the problem step by step. Let me know if you have any further questions!