Question

Answer the following questions about the figure below: (a) What is the value of “x”? (b) What is the value of “y”? (c) What is the value of “z”? (d) What is the value of “t”?

Answer 1

Given that ΔJKL is similar to ΔDEF.

To find the side lengths, we have:

$\frac{DF}{JL}=\frac{DE}{JK}=\frac{EF}{KL}$

Let's solve for y:

$\begin{gathered} \frac{13}{y+8}=\frac{10}{8} \\ \\ \text{Cross multiply:} \\ 10(y+8)=8(13) \\ \\ 10y+80=104 \\ \\ 10y+80-80=104-80 \\ \\ 10y=24 \end{gathered}$

Divide both sides by 10:

$\begin{gathered} \frac{10y}{10}=\frac{24}{10} \\ \\ y=2.4 \end{gathered}$

Let's solve for x:

$\begin{gathered} \frac{x-7}{12}=\frac{10}{8} \\ \\ 8(x-7)=10(12) \\ \\ 8x-56=120 \\ \\ 8x-56+56=120+56 \\ \\ 8x=176 \\ \\ \frac{8x}{8}=\frac{176}{8} \\ \\ x=22 \end{gathered}$

Since they are similar triangles the corresponding angles are congruent.

Thus, we have:

z = 77 degrees

To solve for t, use the triangle angle sum theorem, which states that the sum of interior angles in a triangle is 180 degrees.

Therefore, we have:

t + 77 + 43 = 180

t + 120 = 180

Subtract 120 from both sides:

t + 120 - 120 = 180 - 120

t = 60 degrees

ANSWER:

a) x = 22

b) y = 2.4

c) z = 77°

d) t = 60°