the length of the median drawn to the hypotenuse of a right triangle is represented by 3x-7, while the hypotenuse is represented by 5x-4. Find the length of the median.
If you do not know Thale's Theorem then look it up.
Briefly a right triangle can be inscribed in a circle with the hypotenuse as a diameter and therefore the middle of the hypotenuse as a radius. Then the right angle is also on the circle at the same distance from the center of the circle.
Thus
radius = 3x-7 = (5x-4)/2
6 x - 14 = 5 x - 4
x = 10
3x-7 = 23
http://en.wikipedia.org/wiki/Thales%27_theorem
Shifa
To find the length of the median drawn to the hypotenuse of a right triangle, we need to use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, the hypotenuse is represented by 5x-4, and the median is represented by 3x-7.
So, according to the Pythagorean theorem, we have:
(5x-4)^2 = (3x-7)^2 + (3x-7)^2
Expanding both sides of the equation, we get:
25x^2 - 40x + 16 = 9x^2 - 42x + 49 + 9x^2 - 42x + 49
Combining like terms, we have:
25x^2 - 40x + 16 = 27x^2 - 84x + 98
Bringing all terms to one side of the equation, we get:
0 = 27x^2 - 84x + 98 - 25x^2 + 40x - 16
Simplifying, we have:
0 = 2x^2 - 44x + 82
Now, to solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 2, b = -44, and c = 82. Substituting these values into the formula, we have:
x = (-(-44) ± √((-44)^2 - 4(2)(82))) / (2(2))
Simplifying further, we get:
x = (44 ± √(1936 - 656)) / 4
x = (44 ± √(1280)) / 4
Since we are interested in the length of the median, we need to substitute the value of x back into the expression 3x-7. Let's calculate both possible values of x and find the corresponding lengths of the median.
For x = (44 + √(1280)) / 4:
Length of the median = 3((44 + √(1280)) / 4) - 7
Simplifying, we get:
Length of the median = 33 + 3√(80) - 7
For x = (44 - √(1280)) / 4:
Length of the median = 3((44 - √(1280)) / 4) - 7
Simplifying, we get:
Length of the median = 33 - 3√(80) - 7
So, the length of the median drawn to the hypotenuse of the right triangle can be represented as either 33 + 3√(80) - 7 or 33 - 3√(80) - 7, depending on the value of x.