ANSWER
• x = 12
,• y = 16
,• z = 7
EXPLANATION
Because the triangles are similar, we have that:
• The ratio between corresponding sides is constant:
[tex]\frac{DE}{GH}=\frac{EF}{GF}=\frac{DF}{HF}[/tex]• Corresponding angles are congruent:
[tex]\begin{gathered} \angle D\cong\angle H \\ \angle E\cong\angle G \\ \angle F\cong\angle F \end{gathered}[/tex]We know that the measure of angle E is 16°, so the measure of angle G must be the same because they are congruent,
[tex]16\degree=2(x-4)\degree[/tex]With this equation, we can find x. First, divide both sides by 2,
[tex]\begin{gathered} \frac{16}{2}=\frac{2(x-4)}{2} \\ \\ 8=x-4 \end{gathered}[/tex]And then, add 4 to both sides,
[tex]\begin{gathered} 8+4=x-4+4 \\ \\ 12=x \end{gathered}[/tex]Hence, x = 12.
Now we know that the length of side EF is,
[tex]EF=x-5=12-5=7[/tex]To find y and z, we will use the proportions we got at the top of this explanation,
[tex]\frac{DE}{GH}=\frac{EF}{GF}=\frac{DF}{HF}[/tex]Replace with the known values and the expressions with y and z,
[tex]\frac{25}{6z+8}=\frac{7}{14}=\frac{24}{3y}[/tex]With the first two, we can find z,
[tex]\frac{25}{6z+8}=\frac{7}{14}[/tex]Simplify the right side,
[tex]\frac{25}{6z+8}=\frac{1}{2}[/tex]Rise both sides to the exponent -1 - i.e. flip both sides of the equation,
[tex]\frac{6z+8}{25}=2[/tex]Multiply both sides by 25,
[tex]\begin{gathered} 25\cdot\frac{(6z+8)}{25}=2\cdot25 \\ \\ 6z+8=50 \end{gathered}[/tex]Subtract 8 from both sides,
[tex]\begin{gathered} 6z+8-8=50-8 \\ 6z=42 \end{gathered}[/tex]And divide both sides by 6,
[tex]\begin{gathered} \frac{6z}{6}=\frac{42}{6} \\ \\ z=7 \end{gathered}[/tex]Hence, z = 7.
Finally, with the last two proportions, we can find y,
[tex]\frac{7}{14}=\frac{24}{3y}[/tex]The first two steps are the same we did to find z: simplify the left side and flip both sides,
[tex]2=\frac{3y}{24}[/tex]Multiply both sides by 24,
[tex]\begin{gathered} 24\cdot2=24\cdot\frac{3y}{24} \\ \\ 48=3y \end{gathered}[/tex]And divide both sides by 3,
[tex]\begin{gathered} \frac{48}{3}=\frac{3y}{3} \\ \\ 16=y \end{gathered}[/tex]Hence, y = 16.
