In a right triangle, the length of the longer leg is 7 more inches than the shorter leg. The length of the hypotenuse is 8 more inches than the length of the shorter leg.
A). If the shorter leg is representd by x, write expressions for the longer leg and the hypotenuse in terms of x
B). Write an equation using the Pythagoreum Theorem that relates the three sides together and solve it for the value of x. Then identify the lengths of the longer side and hypotenuse.
B.
(x+7)^2 + x^2 = (x+8)^2
x^2 + 14 x + 49 + x^2 = x^2 + 16 x + 64
x^2 - 2 x - 15 = 0
(x-5)(x+3) = 0
so
x = 5
x + 7 = 12
x+8 = 13
A) Let x represent the length of the shorter leg.
The longer leg would then be x + 7 inches.
The hypotenuse would be x + 8 inches.
B) According to the Pythagorean Theorem, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
Using this, we can write the equation:
x^2 + (x + 7)^2 = (x + 8)^2
Expanding and simplifying this equation, we get:
x^2 + x^2 + 14x + 49 = x^2 + 16x + 64
Combining like terms, we have:
2x^2 + 14x + 49 = x^2 + 16x + 64
Bringing all the terms to one side of the equation, we get:
x^2 - 2x - 15 = 0
Now, we can solve this quadratic equation.
Factoring it, we have:
(x - 5)(x + 3) = 0
Setting each factor to zero, we get:
x - 5 = 0 or x + 3 = 0
Solving for x, we have:
x = 5 or x = -3
Since we are dealing with the lengths of sides, x cannot be a negative value. Therefore, x = 5.
So, the length of the longer leg is 5 + 7 = 12 inches, and the length of the hypotenuse is 5 + 8 = 13 inches.
A). Let x represent the length of the shorter leg.
The longer leg can be represented by x + 7 inches.
The hypotenuse can be represented by x + 8 inches.
B). According to the Pythagorean Theorem, the sum of the squares of the two shorter sides (legs) of a right triangle is equal to the square of the longest side (hypotenuse).
Using this concept, the equation can be written as:
(x)^2 + (x + 7)^2 = (x + 8)^2
Now, let's solve this equation to find the value of x:
x^2 + (x^2 + 14x + 49) = (x^2 + 16x + 64)
x^2 + x^2 + 14x + 49 = x^2 + 16x + 64
2x^2 + 14x + 49 = x^2 + 16x + 64
Rearranging the terms:
2x^2 + 14x + 49 - x^2 - 16x - 64 = 0
x^2 - 2x - 15 = 0
Factoring the quadratic equation:
(x - 5)(x + 3) = 0
Setting each factor equal to zero:
x - 5 = 0 or x + 3 = 0
Solving for x:
x = 5 or x = -3
Since we are dealing with lengths, the value of x cannot be negative. Therefore, the value of x is 5.
The longer leg can be calculated by substituting x = 5 into the expression x + 7:
= 5 + 7
= 12 inches
The hypotenuse can be calculated by substituting x = 5 into the expression x + 8:
= 5 + 8
= 13 inches
So, the longer side (leg) has a length of 12 inches, and the hypotenuse has a length of 13 inches.
A) Let's represent the shorter leg as "x".
The longer leg is 7 more inches than the shorter leg. So, the expression for the longer leg would be "x + 7".
The length of the hypotenuse is 8 more inches than the length of the shorter leg. Thus, the expression for the hypotenuse would be "x + 8".
B) According to the Pythagorean Theorem, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
Using the expressions from part A, the equation would be:
x^2 + (x + 7)^2 = (x + 8)^2
Now, let's solve this equation for the value of x:
Expanding the equation:
x^2 + (x^2 + 14x + 49) = (x^2 + 16x + 64)
Combining like terms:
2x^2 + 14x + 49 = x^2 + 16x + 64
Subtracting x^2 and 16x from both sides:
x^2 - 2x - 15 = 0
Factoring the quadratic equation:
(x - 5)(x + 3) = 0
Setting each factor to zero and solving for x:
x - 5 = 0 -> x = 5
x + 3 = 0 -> x = -3
Since lengths cannot be negative, x = -3 is not a valid solution. Therefore, x = 5.
Now that we have the value of x, we can find the lengths of the longer side and hypotenuse:
The longer leg = x + 7 = 5 + 7 = 12 inches
The hypotenuse = x + 8 = 5 + 8 = 13 inches
So, the lengths of the longer side and hypotenuse are 12 inches and 13 inches, respectively.