GH/HI = 1/3 and HI/IJ = 1/2
If the length of GI is 24 then the length of GJ is ________
First problem: label GH as x. Then HI = 3x. Since GI is 24, you have
x + 3x = 24.
4x = 24,
4x/4 = 24/4
so x = 6.
This means that GH = 6 and HI = 18.
Since HI/JI = 1/2, if HI = 18 then
IJ = 36. The length of GJ is
6 + 18 + 36 = 60,
GJ = 60.
To solve this problem, we can use the transitive property of equality. Since GH/HI = 1/3 and HI/IJ = 1/2, we can combine these two equations together to find the relationship between GH and IJ.
First, we can rewrite GH/HI = 1/3 as GH = (1/3) * HI. Similarly, we can rewrite HI/IJ = 1/2 as HI = (1/2) * IJ.
Now, we can substitute the value of HI from the second equation into the first equation: GH = (1/3) * ((1/2) * IJ). Simplifying this expression gives: GH = (1/6) * IJ.
We are given that the length of GI is 24. Since GH = GI + IH, and given that GH = (1/6) * IJ, we can substitute these values in to find the relationship between GI and IJ.
GI + IH = (1/6) * IJ.
Now, we are given the length of GI, which is 24. Plugging this in to the equation, we have:
24 + IH = (1/6) * IJ.
To find the length of GJ, we need to solve for IJ. Rearranging the equation gives us:
IJ = (6 * (24 + IH)) / 1.
Since HI/IJ = 1/2, we know that IH = (1/2) * IJ. Substituting this value in to the equation, we get:
IJ = (6 * (24 + (1/2) * IJ)) / 1.
To simplify this equation, we can distribute the 6:
IJ = (144 + 3 * IJ) / 1.
Now, we can multiply both sides of the equation by 1 to get rid of the denominator:
IJ * 1 = 144 + 3 * IJ.
Combining like terms, we have:
IJ = 144 + 3IJ.
Subtracting IJ from both sides:
IJ - 3IJ = 144.
Simplifying further:
-2IJ = 144.
Dividing both sides by -2:
IJ = -72.
Since length cannot be negative, it seems there may be an error in the problem or the given information. Assuming there is no error, the length of GJ would be -72, but this is not a valid length.