Using the Intercept Theorem in triangles AMN and ABC, with MN parallel to BC, the value of x is found to be 16. The theorem establishes a proportion, leading to the equation 2x-5 = 27, solving to x = 16.
In a situation where MN is parallel to BC and AM is a transversal, we can apply the similarity criterion known as the Intercept Theorem. According to this theorem, if a pair of parallel lines is intersected by a transversal, the ratio of the lengths of the segments they are divided into is the same for both lines.
In
, we can use the Intercept Theorem to set up the following proportion:
![\[(AM)/(AB) = (AN)/(AC)\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ehl0ianxlwe67zqbimg5d36v0vhrgassfo.png)
Substitute the given values:
![\[(2x-5)/((x+8) + (20)) = (22.5)/(22.5 + 20)\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/k66g72wa33gb4bdpq443i9gvseqrit0o4h.png)
Now, solve for x:
![\[2x-5 = (22.5 \cdot 48)/(42.5) = 27\]](https://img.qammunity.org/2024/formulas/mathematics/high-school/gi6ditjm5hlmaj50yo86gyj12nl8aiho6u.png)
2x = 27 + 5
2x = 32
x = 16
So, the value of x is 16.