The hypotenuse of a right triangle is 1 cm more than twice the shorter leg, and the longer leg is 9 cm less than three times the shorter leg. Find the lengths of the three sides of the triangle..
if the short leg is x,
x^2 + (3x-9)^2 = (2x+1)^2
x = 5/3 or 8
check:
x=5/3, no good; 3x-9 < 0
x=8: triangle is 8-15-17
a well-known Pythagorean triple
Why did the right triangle go to therapy? Because it had hypotenuse issues!
Let's solve this triangle trouble together. Let's assume the length of the shorter leg is x cm. According to the problem, the hypotenuse (the longest side) is 1 cm more than twice the shorter leg. So, the hypotenuse would be 2x + 1 cm.
The problem also states that the longer leg is 9 cm less than three times the shorter leg. So, the longer leg would be 3x - 9 cm.
Now we have all three sides of the triangle:
Shorter leg: x cm
Longer leg: 3x - 9 cm
Hypotenuse: 2x + 1 cm
And there you have it, the lengths of the three sides of the triangle. Remember, though, triangles can be quite complex. So, don't let them triangle you up!
Let's assume the shorter leg of the right triangle is represented by "x" cm.
According to the given information,
The hypotenuse is 1 cm more than twice the shorter leg, so it can be expressed as (2x + 1) cm.
The longer leg is 9 cm less than three times the shorter leg, so it can be expressed as (3x - 9) cm.
Using the Pythagorean theorem, the sum of the squares of the two legs is equal to the square of the hypotenuse:
x^2 + (3x - 9)^2 = (2x + 1)^2
Expanding and simplifying, we get:
x^2 + 9x^2 - 54x + 81 = 4x^2 + 4x + 1
Combining like terms and simplifying further:
10x^2 - 58x + 80 = 0
Now, let's solve this quadratic equation using factoring or the quadratic formula.
Factoring the equation, we have:
2(5x - 8)(x - 10) = 0
Setting each factor equal to zero, we get:
5x - 8 = 0 or x - 10 = 0
Solving for x, we have:
5x = 8 or x = 10
For x = 8/5, it doesn't make sense in this context since we are dealing with lengths in cm, so we can disregard it.
For x = 10, we can substitute this value back into the expressions we found earlier to find the lengths of the sides.
The length of the shorter leg is 10 cm.
The length of the longer leg is (3 * 10) - 9 = 21 cm.
The length of the hypotenuse is (2 * 10) + 1 = 21 cm.
Therefore, the lengths of the three sides of the triangle are: 10 cm, 21 cm, and 21 cm.
To solve this problem, we can call the shorter leg of the right triangle "x".
The longer leg of the triangle is given as 9 cm less than three times the shorter leg. So, the longer leg can be expressed as 3x - 9 cm.
The hypotenuse of the triangle is given as 1 cm more than twice the shorter leg. Therefore, the hypotenuse can be expressed as 2x + 1 cm.
Now, using the Pythagorean theorem, we know that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying this theorem to our triangle, we have the equation:
(2x + 1)^2 = x^2 + (3x - 9)^2
Expanding and simplifying this equation, we get:
4x^2 + 4x + 1 = x^2 + 9x^2 - 54x + 81
Combining like terms, we have:
4x^2 + 4x + 1 = 10x^2 - 54x + 81
Moving all the terms to one side of the equation, we obtain:
6x^2 - 58x + 80 = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Factoring seems to be the easiest approach here.
We can first divide the entire equation by 2 to simplify it:
3x^2 - 29x + 40 = 0
Now, we need to find two numbers that satisfy the equation:
(3x - 5)(x - 8) = 0
From here, we have two possibilities:
1) 3x - 5 = 0
3x = 5
x = 5/3
2) x - 8 = 0
x = 8
Therefore, the possible values for the shorter leg, x, are 5/3 and 8 cm.
Substituting these values back into the original equations, we can find the lengths of the other two sides of the triangle:
For x = 5/3:
Longer leg = 3x - 9 = 3(5/3) - 9 = 5 - 9 = -4 cm (Discard this solution since length cannot be negative)
Hypotenuse = 2x + 1 = 2(5/3) + 1 = 10/3 + 1 = 13/3 cm
For x = 8:
Longer leg = 3x - 9 = 3(8) - 9 = 24 - 9 = 15 cm
Hypotenuse = 2x + 1 = 2(8) + 1 = 16 + 1 = 17 cm
Therefore, the lengths of the sides of the right triangle are:
Shorter leg = 8 cm
Longer leg = 15 cm
Hypotenuse = 17 cm