Given: △acm, m∠c=90°, cp ⊥ am
ac: cm=3: 4, mp-ap=1.

find am.

! i have no idea and have been doing this for hours! i really want to understand brainliest for first correct answer : )

Set up the variables needed

x = AP, y = PM, z = CP, p = AC, q = CM

Based on that, we know x+y = AM

Since MP-AP = 1, we know that y-x = 1 which leads to y = x+1

We will use y = x+1 in a few substitution steps.

The ratio below can help us isolate for p

AC:CM=3:4

AC/CM = 3/4

p/q = 3/4

4p = 3q

p = 3q/4 ... we'll use this in a substitution step

Because we have three similar triangles, we can set up the following ratio and use it to help isolate z^2

(AP)/(PC) = (PC)/(PM)

x/z = z/y

xy = z^2 ... cross multiply

x(x+1) = z^2 ... plug in y = x+1

x^2+x = z^2

z^2 = x^2+x ... we'll use this later in a substitution step

Focus on triangle ACM. Use pythagorean theorem. Isolate q^2

(AC)^2 + (CM)^2 = (AM)^2

(AC)^2 + (CM)^2 = (AP+PM)^2

(p)^2 + (q)^2 = (x+y)^2

(p)^2 + (q)^2 = (x+x+1)^2 ... plug in y = x+1

(p)^2 + (q)^2 = (2x+1)^2

(3q/4)^2 + (q)^2 = (2x+1)^2 ... plug in p = 3q/4

(9q^2)/16 + (q)^2 = (2x+1)^2

(9q^2)/16 + (16q^2)/16 = (2x+1)^2

(9q^2+16q^2)/16 = (2x+1)^2

(25q^2)/16 = (2x+1)^2

25q^2 = 16(2x+1)^2

q^2 = (16(2x+1)^2)/25

q^2 = (16/25)*(2x+1)^2

q^2 = 0.64(2x+1)^2 this will be used later

Move onto triangle APC. Use pythagorean theorem

(AP)^2 + (PC)^2 = (AC)^2

x^2 + z^2 = p^2

x^2 + z^2 = (3q/4)^2 ... plug in p = 3q/4

x^2 + (x^2+x) = (3q/4)^2 ... plug in z^2 = x^2+x

2x^2 + x = (3q/4)^2

2x^2 + x = (9q^2)/(16)

2x^2 + x = (9*0.64(2x+1)^2)/(16) ... plug in q^2 = 0.64(2x+1)^2

2x^2 + x = (5.76(2x+1)^2)/(16)

2x^2 + x = (5.76/16)(2x+1)^2

2x^2 + x = 0.36(2x+1)^2

2x^2 + x = 0.36(4x^2+4x+1)

2x^2 + x = 1.44x^2+1.44x+0.36

2x^2 + x-1.44x^2-1.44x-0.36 = 0

0.56x^2 - 0.44x - 0.36 = 0

(56/100)x^2 - (44/100)x - (36/100) = 0

note how we have an equation with only one variable now

Use the quadratic formula to get this solution set {x = -1/2 = -0.5, x = 9/7 = 1.285714285714}

We will ignore x = -0.5 as a negative length makes no sense

Therefore, x = AP is exactly 9/7 units long

So,

y = x+1

y = 9/7+1

y = (9/7)+(7/7)

y = (9+7)/7

y = 16/7

and therefore

AM = x+y

AM = 9/7+16/7

AM = (9+16)/7

AM = 25/7

AM = 3.57142857142858

The exact length of AM is the fraction 25/7

The approximate length of AM is roughly 3.57142857142858 units.