Okay, this is going to be confusing to describe, so please bear with me (have paper and pencil ready).
There is a right triangle with an altitude of 12. The line of the altitude splits the base into two segments, x and x+7. One leg of the right triangle is y, which is closest to the segment x. The other leg is z, which is closest to segment x+7.
We have to figure out what the values of x, y, and z are. Thanks for helping me out!
Since this is a right triangle, you can use the Pythagorean theorem several times to solve for all unknowns. Let y be the side adjacent to the segment x and z be the side adjacent to x+7. The length of the hypotenuse, c, is 2x + 7.
Then you have
x^2 + 144 = y^2
(x+7)^2 + 144 = z^2
y^2 + z^2 = (2x+7)^2
There are three equations in three unknowns, so with a bit of messy algebra, x, y anx z can all be determined.
Try x=9, y=15 and z = 20. All three triangles are of the 3,4,5 variety
There is a right triangle with an altitude of 12. The line of the altitude splits the base into two segments, x and x+7. One leg of the right triangle is y, which is closest to the segment x. The other leg is z, which is closest to segment x+7.
We have to figure out what the values of x, y, and z are. Thanks for helping me out!
The altitude to the hypotenuse of a right triangle creates two similar triangles, each similar to the original right triangle and to each other.
The altitude to the hypotenuse of a right triangle is the geometric mean between the segments of the hypotenuse created by the point where the altitude intersects the hypotenuse or h^2 = x(x+7).
x/12 = 12/(x + 7) yields x^2 = 7x - 144 = 0
Using the quadratic formula
x [-7+/-sqrt(49 + 576)]/2
x = [-7+/-25]/2 = 18/2 = 9.
Then, y = sqrt(9^2 + 12^2) = 15
Then, z = sqrt(25^2 - 15^2) = 20.
i don't like MATH! MATH MATH MATH!
No problem, I'd be happy to help you solve this problem!
To start, let's draw a right triangle with its altitude of 12. The base of the triangle will be split into two segments, x and x+7. We can label the longest side of the triangle (the hypotenuse) as h, one leg as y, and the other leg as z.
Now, let's use the Pythagorean theorem to solve for h. The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse).
So, we have:
h^2 = y^2 + z^2
Next, let's use the information given in the problem. We know that the altitude of the triangle is 12, which means we can set up the following equation using similar triangles:
y / 12 = x / (x + 7)
Now, we have two equations to work with:
1. h^2 = y^2 + z^2
2. y / 12 = x / (x + 7)
Let's solve for the values of x, y, and z using these equations.
First, let's solve equation 2 for y:
y = (12 * x) / (x + 7)
Now, substitute this into equation 1:
h^2 = ((12 * x) / (x + 7))^2 + z^2
Simplify this equation by squaring both sides and expanding the squared term:
h^2 = (144 * x^2) / (x^2 + 14x + 49) + z^2
Since h^2 is equal to y^2 + z^2, we can rewrite this as:
(144 * x^2) / (x^2 + 14x + 49) + z^2 = ((12 * x) / (x + 7))^2 + z^2
Now, let's solve this equation for x. Multiply both sides of the equation by (x^2 + 14x + 49) to eliminate the denominators:
144 * x^2 + (x^2 + 14x + 49) * z^2 = (12 * x)^2
Expand the right side:
144 * x^2 + x^2 * z^2 + 14x * z^2 + 49 * z^2 = 144 * x^2
Simplify the equation:
x^2 * z^2 + 14x * z^2 + 49 * z^2 = 0
Factor out z^2:
z^2 * (x^2 + 14x + 49) = 0
Since a product is equal to zero if and only if at least one of the factors is zero, we have two possible cases:
1. z^2 = 0
In this case, z must be equal to zero.
2. x^2 + 14x + 49 = 0
We can solve this quadratic equation by factoring:
(x + 7)(x + 7) = 0
(x + 7)^2 = 0
To solve for x, we take the square root of both sides:
x + 7 = 0
x = -7
So, we have two possible values for x: x = -7 or z = 0.
Now, let's substitute these values back into equation 2 to solve for y:
For x = -7:
y = (12 * (-7)) / (-7 + 7) = 0
So, when x = -7, y = 0.
For z = 0:
y = (12 * x) / (x + 7) = (0 * x) / (x + 7) = 0
So, when z = 0, y = 0.
In conclusion, we find that when one leg of the right triangle is closest to segment x, and the other leg is closest to segment x+7, the values of x, y, and z are: x = -7, y = 0, and z = 0.
I hope this explanation helps! If you have any further questions, feel free to ask.