Part E Determine $$ \sin (m \angle D) $$ and justify your reasoning. Drag the correct equations into the boxes to correctly model the situation. Show Hints The sines of corresponding angles are equivalent because the scale factor will cancel out: ! $$ \sin D=\frac{24}{53 k}=\frac{28}{83} $$ $$ \sin D=\sin A=\frac{45}{35} $$ $$ \sin D=\frac{44}{64}=\frac{45}{65} $$ $$ \sin D=\sin A=\frac{2 \pi}{53} $$

Question

UpStudy AI · Accepted Answer

To determine $$ \sin (m \angle D) $$ and correctly model the situation, we'll analyze the given options and justify the appropriate choice based on the properties of similar triangles and the behavior of trigonometric ratios.

### Correct Equation:
$$
\sin D = \frac{24}{53k} = \frac{28}{83}
$$

### Justification:
1. **Similar Triangles and Scale Factor:**
   - In similar triangles, corresponding angles are equal, and the ratios of corresponding sides are proportional.
   - Let’s denote the scale factor between the two triangles as $$ k $$. This means that each side of one triangle is $$ k $$ times the corresponding side of the other triangle.

2. **Sine of Corresponding Angles:**
   - The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
   - Since the triangles are similar, the sine of the corresponding angles remains the same because the scale factor $$ k $$ cancels out.
   - Therefore, 
     $$
     \sin D = \sin A
     $$
     where $$ \angle D $$ in one triangle corresponds to $$ \angle A $$ in the similar triangle.

3. **Applying the Given Ratios:**
   - From the equation:
     $$
     \sin D = \frac{24}{53k} = \frac{28}{83}
     $$
   - This shows that the sine of angle $$ D $$ is equal to the ratio $$ \frac{28}{83} $$ after the scale factor $$ k $$ cancels out, confirming that the corresponding sines are indeed equal.

### Why Other Options Are Incorrect:
- **$$ \sin D = \sin A = \frac{45}{35} $$:**
  - The sine of an angle cannot exceed 1, and $$ \frac{45}{35} $$ is greater than 1, which is not possible for a sine value.
  
- **$$ \sin D = \frac{44}{64} = \frac{45}{65} $$:**
  - These ratios are inconsistent and do not simplify to the same value, which contradicts the property that corresponding sines in similar triangles are equal.
  
- **$$ \sin D = \sin A = \frac{2 \pi}{53} $$:**
  - The sine of an angle is a dimensionless ratio and should not involve $$ \pi $$ in this context. Additionally, $$ \frac{2 \pi}{53} $$ is an angle measure, not a sine value.

### Conclusion:
The correct equation to model the situation is:
$$
\sin D = \frac{24}{53k} = \frac{28}{83}
$$
This equation correctly represents the relationship between the sines of corresponding angles in similar triangles, with the scale factor $$ k $$ appropriately canceling out.
The correct equation to determine $$ \sin (m \angle D) $$ is: $$ \sin D = \frac{24}{53k} = \frac{28}{83} $$ This equation shows that the sine of angle $$ D $$ is equal to $$ \frac{28}{83} $$ after accounting for the scale factor $$ k $$ between the similar triangles.

Part E
Determine and justify your reasoning.
Drag the correct equations into the boxes to correctly model the situation.

Show Hints

The sines of corresponding angles are equivalent because the scale factor will
cancel out: !

Upstudy AI Solution

Answer

Solution

Correct Equation:

Justification:

Why Other Options Are Incorrect:

Conclusion:

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Part E Determine and justify your reasoning. Drag the correct equations into the boxes to correctly model the situation. Show Hints The sines of corresponding angles are equivalent because the scale factor will cancel out: !