the hypotenuse of a right triangle is 3 less than 4 times the shortest leg. if the longest leg is 24cm, how long is the hypotenuse
Some basics:
24 = longest leg
x = shortest leg
4x - 3 = hypotenuse
Remember the following:
a^2 + b^2 = c^2
24^2 + x^2 = (4x - 3)^2
576 + x^2 = 16x^2 - 24x + 9
Set the equation equal to 0:
0 = 15x^2 - 24x - 567
Factoring:
0 = 3(5x^2 - 8x - 189)
0 = 3(5x + 27)(x - 7)
Set the factors equal to 0:
5x + 27 = 0; x = -27/5
x - 7 = 0; x = 7
Since you can't have a negative number, x = 7 is your only answer.
I hope this explanation will help with other problems of this type.
Remember to substitute 7 for x to figure out the length of the hypotenuse!
Let's assume that the shortest leg of the right triangle is represented by the variable "x".
Given:
Longest leg = 24 cm
Hypotenuse = 3 less than 4 times the shortest leg
This can be represented as an equation:
Hypotenuse = 4x - 3
We are given that the longest leg is 24 cm. So, we can substitute this information into the equation:
24 = 4x - 3
To solve for x, we'll isolate it on one side of the equation by adding 3 to both sides:
24 + 3 = 4x
27 = 4x
To find the value of x, we divide both sides by 4:
27/4 = x
Now, we can calculate the value of x:
x = 6.75
Therefore, the length of the hypotenuse is 4 times the shortest leg (4 * 6.75) minus 3, which is:
Hypotenuse = (4 * 6.75) - 3
Hypotenuse = 27 - 3
Hypotenuse = 24 cm.
So, the length of the hypotenuse is 24 cm.
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's denote the shortest leg as "x" and the hypotenuse as "y". The longest leg is given as 24 cm.
According to the information given, we have the following relationship:
y = 4x - 3 (the hypotenuse is 3 less than 4 times the shortest leg)
Using the Pythagorean theorem, we can write the equation as:
x^2 + 24^2 = y^2
Substituting the value of y from the first equation into the second equation, we have:
x^2 + 24^2 = (4x - 3)^2
Expanding and simplifying the equation:
x^2 + 576 = 16x^2 - 24x + 9
Combining like terms and transferring all terms to one side:
0 = 15x^2 - 24x - 567
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. After finding the value(s) of x, we can substitute it back into the equation y = 4x - 3 to find the length of the hypotenuse.
Note: The answer may include two possible lengths of the hypotenuse since quadratic equations can have two real solutions.