Question

I am trying to solve this triangle for x y and z and I am really struggling here is the image. I hope u can offer some help. Thank you!

Answer 1

Given the triangle in the figure shown;

To determine: x, y, and z

Solution:

Let us labelled the diagram as below:

Compairing the given diagram to the labelled diagram above. It can be observed that

`\begin{gathered} AB=6z+2 \\ BC=56 \\ AC=4x \end{gathered}`

Using the triagle theorem below:

i) If a line intersecting two sides of a triangle is parallel to the remaining side, then the smaller triangle created by the parallel line is similar to the larger, original triangle.

ii) The side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally.

iii) The side splitter theorem can be extended to include parallel lines that lie outside a triangle. If a line lying outside a triangle is parallel to one side of the triangle and intersects the extensions of the other two sides of the triangle, then the line divides the extensions of those sides proportionally.

iv) The converse of the side splitter theorem states that if a line splits two sides of a triangle proportionally, then that line is parallel to the remaining side,

v) If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally

It can be observed that AB is parallel to DF, then AB is twice of DF

`\begin{gathered} AB=2* DF \\ 6z+2=2*40 \\ 6z+2=80 \\ 6z=80-2 \\ 6z=78 \\ z=(78)/(6) \\ z=13 \end{gathered}`

Also, ED is parallel to BC, then BC is twice ED

`\begin{gathered} BC=2* ED \\ 56=2*4y \\ 56=8y \\ 8y=56 \\ y=(56)/(8) \\ y=7 \end{gathered}`

Also, EF is parallel to AC, then AC is twice EF

`\begin{gathered} 4x=2*30 \\ 4x=60 \\ x=(60)/(4) \\ x=15 \end{gathered}`

Hence:

x = 15

y = 7

z = 13