BAC= 90 dgrees. The lenght of the three sides are xcm, (x=7) 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 part (a)
c. Write down the value of the perimeter of the triangle.
If you do not say which side is the hypotenuse (opposite the right angle), I can not help you.
hypotenuse^2 = side^2 + other side^2
x^2 + (x+7)^2 = (x+8)^2
solve for x, or save yourself some time by memorizing a few basic tripes, such as 5-12-13
a. To write down the quadratic equation that links the three sides of the triangle, let's use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c and angle C opposite side c, the following relationship holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, we are given that the angle BAC is 90 degrees, so angle C is opposite the side of length x + 8 cm. Let's denote the sides as follows:
a = x cm
b = 7 cm
c = x + 8 cm
C = 90 degrees
Using the Law of Cosines, we can write the quadratic equation:
(x + 8)^2 = x^2 + 7^2 - 2x * 7 * cos(90)
Simplifying this equation will give us the quadratic equation linking the three sides.
b. To solve the quadratic equation, we expand and simplify it:
x^2 + 16x + 64 = x^2 + 49 - 14x
Rearranging the terms and combining like terms:
16x + 14x = 49 - 64
30x = -15
Dividing both sides of the equation by 30:
x = -15/30
Simplifying further:
x = -1/2
Therefore, the solution to the quadratic equation is x = -1/2.
c. To find the perimeter of the triangle, we need to add up the lengths of all three sides. The sides are given as follows:
Side AB = x cm
Side BC = 7 cm
Side AC = x + 8 cm
Substituting the value of x we found in part (b), which is x = -1/2, into the equations above, we can calculate the lengths of the sides:
Side AB = -1/2 cm
Side BC = 7 cm
Side AC = (-1/2 + 8) cm = 7.5 cm
Now we can calculate the perimeter of the triangle by adding the lengths of all three sides:
Perimeter = Side AB + Side BC + Side AC
Perimeter = (-1/2) cm + 7 cm + 7.5 cm
Perimeter = 13 cm + 7.5 cm
Perimeter = 20.5 cm
Therefore, the value of the perimeter of the triangle is 20.5 cm.