Calculating the Length of GH in a Right Triangle
To solve for GH, we will use the properties of right triangles and the Pythagorean theorem. From the image, it looks like we have a right triangle with GE as the hypotenuse, and segments GH and ED as the other two sides.
We are given:
- ED = 3 units (vertical leg)
- GD = 13 units (horizontal leg)
Using the Pythagorean theorem (a^2 + b^2 = c^2) for right triangle FED, where a and b are the legs and c is the hypotenuse (FE in this case), we can calculate the length of FE.
First, we find the length of GD by subtracting the length of HD from GH:
GH = GD - HD
Since GD = 13 and HD = 9 (from the information given in the image),
GH = 13 - 9 = 4 units
Now we have the lengths of both legs of triangle FED:
GD = 13 units
ED = 3 units
We can now find the hypotenuse FE using the Pythagorean theorem:
FE^2 = GD^2 + ED^2
FE^2 = 13^2 + 3^2
FE^2 = 169 + 9
FE^2 = 178
FE = √178
Since GE is the diameter of the circle, which is FE here, and G to H is the radius, GH is half of GE:
GH = FE / 2
GH = √178 / 2
This is the length of GH in terms of the square root. If a decimal value is needed, we would calculate the square root of 178 and then divide by 2 to find GH. However, the value appears to be a radical rather than a decimal, so GH = √178 / 2.
