Question

evaluate independent practice
lesson 12.1 homework
complete problems 1-6 below for independent practice.
when you are finished, check the solutions with your teacher. figures are not drawn to scale. you can assume that all segments that appear to be parallel are indeed parallel.

1. find the missing length. the triangles in each pair are similar.
2. find the missing length. the triangles in each pair are similar.
3. find the missing length. the triangles in each pair are similar.
4. \\(ts \parallel lk\\). (solve for \\(lt\\). then, using that value, solve for \\(tj\\)).
5. \\(cb \parallel qr\\). solve for \\(ur\\). then, using that value, solve for \\(sr\\).
6. \\(nm \parallel ab\\). solve for \\(an\\). then, using that value, solve for \\(ac\\).

Explanation:

Verify parallel lines

Using the Proportional Segments in Triangles knowledge point, we check if \(\frac{VT}{TR} = \frac{VU}{US}\).

$$ \frac{VT}{TR} = \frac{90}{72} = 1.25 $$

$$ \frac{VU}{US} = \frac{67.5}{54} = 1.25 $$

Since the ratios are equal, \(\overline{TU} \parallel \overline{RS}\).

Solve Question 1

Using the Proportional Segments in Triangles knowledge point, we set up the ratio for similar triangles:

$$ \frac{x}{33} = \frac{54}{36} $$

Using the Solving Linear Equations knowledge point:

$$ x = 33 \cdot 1.5 = 49.5 $$

Solve Question 2

Using the Proportional Segments in Triangles knowledge point, we set up the ratio:

$$ \frac{x}{24} = \frac{70}{42} $$

Using the Solving Linear Equations knowledge point:

$$ x = 24 \cdot \frac{5}{3} = 40 $$

Solve Question 3

Using the Proportional Segments in Triangles knowledge point, we set up the ratio:

$$ \frac{x}{10} = \frac{7}{5} $$

Using the Solving Linear Equations knowledge point:

$$ x = 10 \cdot 1.4 = 14 $$

Solve Question 4

Using the Proportional Segments in Triangles knowledge point, we first solve for \(LT\):

$$ \frac{LT}{35} = \frac{30}{70} \implies LT = 35 \cdot \frac{3}{7} = 15 $$

Then, we find \(TJ\):

$$ TJ = LJ - LT = 35 - 15 = 20 $$

Solve Question 5

Using the Proportional Segments in Triangles knowledge point, we first solve for \(UR\):

$$ \frac{UR}{30} = \frac{56}{35} \implies UR = 30 \cdot 1.6 = 48 $$

Then, we find \(SR\):

$$ SR = UR - US = 48 - 30 = 18 $$

Solve Question 6

Using the Proportional Segments in Triangles knowledge point, we first solve for \(AN\):

$$ \frac{AN}{10} = \frac{70}{28} \implies AN = 10 \cdot 2.5 = 25 $$

Then, we find \(AC\):

$$ AC = AN + NC = 25 + 10 = 35 $$

Answer:

Question 1 (Your Turn)

Since \(\frac{VT}{TR} = \frac{VU}{US} = 1.25\), by the Converse of the Triangle Proportionality Theorem, \(\overline{TU} \parallel \overline{RS}\).

Homework Questions

No.	Answer
2	\(x = 40\)
3	\(x = 14\)
4	\(LT = 15\), \(TJ = 20\)
5	\(UR = 48\), \(SR = 18\)
6	\(AN = 25\), \(AC = 35\)