Question

I need help with #1 of this problem. It has writings on it because I just looked up the answer because I’m confused but I want to know the answer and how to do it with work provided please

Answer 1

In the figure below

1) Using the theorem of similar triangles (ΔBXY and ΔBAC),

`(BX)/(BA)=(BY)/(BC)=(XY)/(AC)`

Where

`\begin{gathered} BX=4 \\ BA=5 \\ BY=6 \\ BC\text{ = x} \end{gathered}`

Thus,

`\begin{gathered} (4)/(5)=(6)/(x) \\ \text{cross}-\text{multiply} \\ 4* x=6*5 \\ 4x=30 \\ \text{divide both sides by the coefficient of x, which is 4} \\ \text{thus,} \\ (4x)/(4)=(30)/(4) \\ x=7.5 \end{gathered}`

thus, BC = 7.5

2) BX = 9, BA = 15, BY = 15, YC = y

In the above diagram,

`\begin{gathered} BC=BY+YC \\ \Rightarrow BC=15\text{ + y} \end{gathered}`

Thus, from the theorem of similar triangles,

`\begin{gathered} (BX)/(BA)=(BY)/(BC)=(XY)/(AC) \\ (9)/(15)=(15)/(15+y) \end{gathered}`

solving for y, we have

thus, YC = 10.