BAC=90 degrees. The length of the three sides are x cm, (x+7) cm, and (x+8) cm.
a. Write down and simplify a quadratic equation in x which links the three sides of the triangle.
b. Solve the quadratic equation found in (a)
c. Write the value of the perimeter of the triangle.
a. x^2 + (x + 7)^2 = (x + 8)^2 ... 2 x^2 + 14 x + 49 = x^2 + 16 x + 64
b. x^2 - 2x - 15 = 0 ... (x - 5) (x + 3) = 0 ... x = 5 , x = -3
... the negative result is not realistic
c. a 5-12-13 Pythagorean triple ... perimeter = 30
a. Sure! Let's start with the Pythagorean Theorem. In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. So, we can write the equation using these sides:
(x + 7)^2 + x^2 = (x + 8)^2
Now, let's simplify it a bit:
x^2 + 14x + 49 + x^2 = x^2 + 16x + 64
Combining like terms:
2x^2 + 14x + 49 = x^2 + 16x + 64
b. To solve the quadratic equation, let's rearrange it:
2x^2 + 14x + 49 - x^2 - 16x - 64 = 0
Combining like terms again:
x^2 - 2x - 15 = 0
Now, we can factor this quadratic equation:
(x - 5)(x + 3) = 0
So, the solutions are x = 5 and x = -3.
c. To find the perimeter of the triangle, we need to add up the lengths of all three sides:
Perimeter = x + (x + 7) + (x + 8)
Now, substituting the values we found in part b:
Perimeter = 5 + (5 + 7) + (5 + 8)
Perimeter = 5 + 12 + 13
Perimeter = 30 cm
So, the value of the perimeter of the triangle is 30 cm.
a. To find the quadratic equation that links the three sides of the triangle, we will use the cosine law formula. According to the cosine law,
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, a=x cm, b=(x+7) cm, and C=90 degrees. So, we can write the equation as:
(x+8)^2 = x^2 + (x+7)^2 - 2x(x+7)*cos(90)
Expanding the equation and simplifying:
x^2 + 16x + 64 = x^2 + x^2 + 14x + 49 - 2x^2 - 14x
Combining like terms:
x^2 - 2x^2 + x^2 + 16x - 14x + 14x + 64 - 49 = 0
Simplifying further:
-2x^2 + 16x + 15 = 0
This is the quadratic equation that links the three sides of the triangle.
b. To solve the quadratic equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -2, b = 16, and c = 15. Substituting these values into the formula:
x = (-(16) ± √((16)^2 - 4(-2)(15))) / (2(-2))
Simplifying inside the square root:
x = (-16 ± √(256 + 120)) / (-4)
x = (-16 ± √376) / (-4)
x = (-16 ± 19.39) / (-4)
So, we have two possible values for x:
x1 = (-16 + 19.39) / (-4) = 0.8475
x2 = (-16 - 19.39) / (-4) = 8.8475
c. The perimeter of the triangle can be calculated by adding the lengths of all three sides:
Perimeter = x + (x + 7) + (x + 8)
Substituting the value of x = 0.8475:
Perimeter = 0.8475 + (0.8475 + 7) + (0.8475 + 8)
Perimeter = 0.8475 + 7.8475 + 8.8475
Perimeter = 17.5425 cm
Therefore, the value of the perimeter of the triangle is 17.5425 cm.
To solve this problem, we will first use the properties of a triangle to set up a quadratic equation in terms of x. Then we will solve the equation to find the value(s) of x. Finally, we will use those values to calculate the perimeter of the triangle.
a. Writing down the quadratic equation in x:
In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. So for our triangle, using the given lengths:
x + (x+7) > (x+8)
2x + 7 > x + 8
2x - x > 8 - 7
x > 1
Since we need to find the quadratic equation, we'll square both sides of the above inequality:
(x > 1) → (x^2 > 1^2)
x^2 > 1
So, the quadratic equation that links the three sides of the triangle is:
x^2 - 1 = 0
b. Solving the quadratic equation:
To solve the equation x^2 - 1 = 0, we can factor it or use the quadratic formula.
Since this equation is simple, we can factor it as:
(x + 1)(x - 1) = 0
Setting each factor equal to zero, we get:
x + 1 = 0 or x - 1 = 0
Solving for x, we find two solutions:
x = -1 or x = 1
c. Finding the perimeter of the triangle:
Now that we have the values of x, we can substitute them into the expression for the length of the sides (x cm, (x+7) cm, and (x+8) cm) to find the lengths of each side.
For x = -1:
Side 1 = x cm = -1 cm
Side 2 = (x+7) cm = (-1+7) cm = 6 cm
Side 3 = (x+8) cm = (-1+8) cm = 7 cm
For x = 1:
Side 1 = x cm = 1 cm
Side 2 = (x+7) cm = (1+7) cm = 8 cm
Side 3 = (x+8) cm = (1+8) cm = 9 cm
We now have two possible triangles with different side lengths. To determine the perimeter, we add up the lengths of the sides for each triangle:
Perimeter of Triangle 1 = Side 1 + Side 2 + Side 3
= -1 cm + 6 cm + 7 cm
= 12 cm
Perimeter of Triangle 2 = Side 1 + Side 2 + Side 3
= 1 cm + 8 cm + 9 cm
= 18 cm
So, the perimeters of the two possible triangles are 12 cm and 18 cm.