Answer: The answer is [tex]3\dfrac{4}{7}.[/tex]
Step-by-step explanation: Given in the question that ΔAM is a right-angled triangle, where ∠C = 90°, CP ⊥ AM, AC : CM = 3 : 4 and MP - AP = 1. We are to find AM.
Let, AC = 3x and CM = 4x.
In the right-angled triangle ACM, we have
[tex]AM^2=AC^2+CM^2=(3x)^2+(4x)^2=9x^2+16x^2=25x^2\\\\\Rightarrow AM=5x.[/tex]
Now,
[tex]AM=AP+PM=AP+(AP+1)\\\\\Rightarrow 2AP=AM-1\\\\\Rightarrow 2AP=5x-1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(A)[/tex]
Now, since CP ⊥ AM, so ΔACP and ΔMCP are both right-angled triangles.
So,
[tex]CP^2=AC^2-AP^2=CM^2-MP^2\\\\\Rightarrow (3x)^2-AP^2=(4x)^2-(AP+1)^2\\\\\Rightarrow 9x^2-AP^2=16x^2-AP^2-2AP-1\\\\\Rightarrow 2AP=7x^2-1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(B)[/tex]
Comparing equations (A) and (B), we have
[tex]5x-1=7x^2-1\\\\\Rightarrow 5x=7x^2\\\\\Rightarrow x=\dfrac{5}{7},~\textup{since }x\neq 0.[/tex]
Thus,
[tex]AM=5\times\dfrac{5}{7}=\dfrac{25}{7}=3\dfrac{4}{7}.[/tex]
