A toy manufacturer needs a piece of plastic in the shape of a right triangle with the longer leg 5cm more than the shorter leg and the hypotenuse 10 cm more than the shorter leg. How long should the sides of the triangles be?
Let the lengths of the legs be a, b and c in increasing order. c is the hypotenuse.
b = a + 5
c = a + 10
c^2 = a^2 + b^2
Three equations in three unknowns. See what you can do with that.
c^2 = a^2 + a^2 + 10a + 25
= 2a^2 + 10a + 25
c^2 = a^2 + 20a + 100
Set the two last equations for c^2 equal to each other and solve for a.
a^2 -10a -75 = 0
(a-15)(a+5) = 0
Only the positive root is allowed.
a = 15
b = 20
c = 25
Why did the triangle go on a diet?
Because it wanted to be a right angle!
To solve the problem, let's call the shorter leg x cm. According to the information given, the longer leg would be x + 5 cm, and the hypotenuse would be x + 10 cm. Since it's a right triangle, we can use the Pythagorean theorem to find the value of x. We have:
(x^2) + (x + 5)^2 = (x + 10)^2
Simplifying the equation, we get:
x^2 + (x^2 + 10x + 25) = x^2 + 20x + 100
Combining like terms, we have:
2x^2 + 10x + 25 = x^2 + 20x + 100
Bringing everything to one side of the equation, we get:
x^2 - 10x - 75 = 0
Using the quadratic formula, we find that x ≈ 12.46 cm.
So, the shorter leg of the triangle would be approximately 12.46 cm, the longer leg would be 17.46 cm (12.46 + 5), and the hypotenuse would be 22.46 cm (12.46 + 10).
Let's start by representing the shorter leg of the right triangle as x cm.
According to the given information:
1. The longer leg is 5 cm more than the shorter leg, so its length will be x + 5 cm.
2. The hypotenuse is 10 cm more than the shorter leg, so its length will be x + 10 cm.
Now, we can use the Pythagorean theorem to find the value of x.
According to the Pythagorean theorem:
(shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2
Substituting the values we have:
x^2 + (x + 5)^2 = (x + 10)^2
Expanding the equation:
x^2 + (x^2 + 10x + 25) = x^2 + 20x + 100
Simplifying the equation:
x^2 + x^2 + 10x + 25 = x^2 + 20x + 100
Combining like terms:
2x^2 + 10x + 25 = x^2 + 20x + 100
Rearranging the equation:
2x^2 - x^2 + 10x - 20x = 100 - 25
Simplifying the equation:
x^2 - 10x = 75
Moving all terms to one side:
x^2 - 10x - 75 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
By factoring:
(x - 15)(x + 5) = 0
Setting each factor to zero:
x - 15 = 0 or x + 5 = 0
Solving for x:
x = 15 or x = -5
Since lengths cannot be negative, we discard the solution x = -5.
So, the shorter leg of the right triangle should be x = 15 cm.
Now, we can substitute x = 15 into the given information to find the lengths of the other sides:
Shorter leg = x = 15 cm
Longer leg = x + 5 = 15 + 5 = 20 cm
Hypotenuse = x + 10 = 15 + 10 = 25 cm
Therefore, the sides of the right triangle should be:
Shorter leg = 15 cm,
Longer leg = 20 cm,
Hypotenuse = 25 cm.
To find the lengths of the sides of the right triangle, let's assume that the shorter leg is represented by the variable "x" cm.
According to the given information:
- The longer leg is 5 cm more than the shorter leg, which means the length of the longer leg would be x + 5 cm.
- The hypotenuse is 10 cm more than the shorter leg, so the hypotenuse would have a length of x + 10 cm.
Using the Pythagorean Theorem, we can calculate the sides of the triangle. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the two other sides.
So, applying the Pythagorean theorem, we can write the equation as:
(x^2) + ((x + 5)^2) = ((x + 10)^2)
Let's solve this equation to find the value of "x".
Expanding and simplifying the equation, we have:
x^2 + (x^2 + 10x + 25) = x^2 + 20x + 100
Combining the like terms, we have:
2x^2 + 10x + 25 = x^2 + 20x + 100
Move all the terms to one side of the equation:
2x^2 + 10x + 25 - x^2 - 20x - 100 = 0
Combine like terms again:
x^2 - 10x - 75 = 0
Now, we need to solve this quadratic equation.
Factor the equation:
(x - 15)(x + 5) = 0
Setting each factor equal to zero:
x - 15 = 0 ---> x = 15
x + 5 = 0 ---> x = -5
Since the length of a side cannot be negative, we discard the negative solution.
Thus, the shorter leg of the right triangle should be 15 cm.
The longer leg would then be x + 5 = 15 + 5 = 20 cm.
And the hypotenuse would be x + 10 = 15 + 10 = 25 cm.
Therefore, the sides of the right triangle should be 15 cm, 20 cm, and 25 cm.