Question

PLEASE HELP 45 POINTS!!!!!

Answer 1

We see 2 congruent triangles: Triangle DBC and FEC. Let's set up a proportion to find the value of x and length of DF.

`(BE+EC)/(DF+FC) =(EC)/(FC)`

`(3+4)/((x+7)+16) =(4)/(16)`

Cross multiply and solve for x.

`(3+4)*16=((x+7)+16)*4`

`7*16=(x+23)*4`

`112=(x+23)*4`

`28= (x+23)`

x= 5

Now to solve for the length of DF, plug in x into DF.

DF=(x+7)

DF=(5+7)

DF= 12

Answer 2

• 
  `\sf x = 5`

• 
  `\sf DF = 12`

setting up proportionality equation:

`\sf (big-side)/(big-diagonal) = \sf (small-side)/(small-diagonal)`

`\sf \hookrightarrow (7)/(16+x+7) = (4)/(16)`

`\sf \hookrightarrow (7)/(23+x) = (4)/(16)`

`\sf \hookrightarrow 4(23+x) = 112`

`\sf \hookrightarrow 92 + 4x = 112`

`\sf \hookrightarrow 4x = 112- 92`

`\sf \hookrightarrow 4x = 20`

`\sf \hookrightarrow x = 5`

Find DF:

`\hookrightarrow \sf x + 7`

`\hookrightarrow \sf 5 + 7`

`\hookrightarrow \sf 12`