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lesson 11.3 checkpoint once you have completed the above problems and c…

Question

lesson 11.3 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

in the diagram, \\(\delta npq \sim \delta nlm\\) and \\(nl = 3\\)

  1. find the value of \\(x\\).
  2. find the value of \\(y\\). remember that \\(nl = 3\\)
  3. what must the missing length be for the two figures to be similar?

Explanation:

Identify corresponding angles for Question 1

Using the Corresponding Angles of Similar Figures knowledge point
Given \(\Delta NPQ \sim \Delta NLM\), the corresponding angles are:

$$ \angle NPQ \cong \angle NLM \implies m\angle NPQ = m\angle NLM $$

From the diagram:

$$ m\angle NPQ = 60^\circ,\quad m\angle NLM = (3x + 18)^\circ $$

Solve for x

Using the Solving Linear Equations knowledge point

$$ LATEXBLOCK0 $$

Set up the ratio for Question 2

Using the Corresponding Sides of Similar Figures knowledge point
The similarity statement \(\Delta NPQ \sim \Delta NLM\) gives the ratio of corresponding sides:

$$ \frac{NP}{NL} = \frac{NQ}{NM} $$

From the diagram and given values:

$$ NP = y,\quad NL = 3,\quad NQ = 3.2\text{ cm},\quad NM = 4\text{ cm} $$

Solve for y

Using the Solving Linear Equations knowledge point

$$ LATEXBLOCK1 $$

Determine the missing length for Question 3

Using the Corresponding Sides of Similar Figures knowledge point
Let the missing side of the first quadrilateral be \(z\). Comparing corresponding sides of the two similar quadrilaterals:

$$ \frac{z}{28} = \frac{9}{24} = \frac{25}{20} \implies \text{No, check orientation.} $$

Matching corresponding sides by ratio:

$$ \frac{\text{Top-right}}{\text{Top-right}} = \frac{9}{24} = \frac{3}{8} $$
$$ \frac{\text{Bottom}}{\text{Bottom}} = \frac{25}{20} = \frac{5}{4} \implies \text{Orientation is rotated/reflected.} $$

Let's pair the sides by ordering them from smallest to largest:
First figure: \(9 < z < 25\)
Second figure: \(20 < 24 < 28\)
Matching ratios:

$$ \frac{9}{20} eq \frac{25}{28} $$

Let's match corresponding sides of the similar quadrilaterals:

$$ \frac{\text{Side}_1}{\text{Side}_2} = k $$

If the scale factor is \(k = \frac{20}{25} = 0.8\):

$$ \frac{24}{z} = 0.8 \implies z = 30 \quad (\text{not matching } 28 \text{ and } 9) $$

If the scale factor is \(k = \frac{24}{25}\):
Let's match the corresponding sides directly:

$$ \frac{24}{9} = \frac{8}{3} \implies \text{Missing side } z \text{ corresponds to } 20 \text{ or } 28. $$

Using the ratio of known corresponding sides:

$$ \frac{24}{9} = \frac{8}{3} \approx 2.67 $$
$$ \frac{28}{z} = \frac{8}{3} \implies z = \frac{84}{8} = 10.5 $$
$$ \frac{20}{25} = 0.8 \implies \text{Alternative orientation:} $$

Let the vertices be matched. The sides of the first figure are \(9\), \(25\), and \(z\). The sides of the second figure are \(24\), \(20\), and \(28\).
Matching the ratios:

$$ \frac{24}{z} = \frac{20}{25} = \frac{28}{9} \implies \text{Not consistent.} $$

Let's match:

$$ \frac{24}{25} \approx 0.96,\quad \frac{20}{z} = \frac{24}{25} \implies z \approx 20.8 $$

Let's match:

$$ \frac{28}{25} = 1.12,\quad \frac{24}{z} = 1.12 \implies z = 21.4 $$

Let's check:
\[
\frac{28}{25} = \frac{24}{z}…

Answer:

Question 1

\(x = 14\)

Question 2

\(y = 2.4\)

Question 3

\(35\)