Question
Triangle VXW is similar to triangle ZXY.
Triangle V X W. Side V W is 12 miles and side V X is 14 miles. Triangle Z X Y. Side Y Z is 72 miles and side X Z is x miles.
Kara finds the value of x. Her work is shown below.
Step 1 StartFraction 14 Over 72 EndFraction = StartFraction 12 Over x EndFraction
Step 2 14 x = (72) (12)
Step 3 14 x = 864
Step 4 x = 61.7
What is Kara’s first error?
A. Kara should have written the proportion in step 1 as StartFraction 12 Over 72 EndFraction = StartFraction 14 Over x EndFraction.
B. Kara should have written the cross-product in step 2 as 12 x = (72) (14).
C. Kara did not correctly divide 864 by 14 going from step 3 to step 4.
D. Kara did not find the correct product of 72 and 12 going from step 2 to step 3.
Triangle V X W. Side V W is 12 miles and side V X is 14 miles. Triangle Z X Y. Side Y Z is 72 miles and side X Z is x miles.
Kara finds the value of x. Her work is shown below.
Step 1 StartFraction 14 Over 72 EndFraction = StartFraction 12 Over x EndFraction
Step 2 14 x = (72) (12)
Step 3 14 x = 864
Step 4 x = 61.7
What is Kara’s first error?
A. Kara should have written the proportion in step 1 as StartFraction 12 Over 72 EndFraction = StartFraction 14 Over x EndFraction.
B. Kara should have written the cross-product in step 2 as 12 x = (72) (14).
C. Kara did not correctly divide 864 by 14 going from step 3 to step 4.
D. Kara did not find the correct product of 72 and 12 going from step 2 to step 3.
Answers
GPT-4o mini
To determine Kara's first error, let's analyze her steps one by one.
In similarity of triangles, the sides correspond to each other based on the established ratios. For triangles VXW and ZXY, we can set up the proportion based on the corresponding sides.
1. **Step 1: Kara writes the proportion as \(\frac{14}{72} = \frac{12}{x}\).**
- The sides WX and XY (which is 12 and \(x\)) correspond to each other. However, she should have written it as \(\frac{12}{72} = \frac{14}{x}\), because the relationship in triangle similarity is based on the corresponding sides.
- Therefore, **A** is correct: Kara should have written the proportion in step 1 as \(\frac{12}{72} = \frac{14}{x}\).
2. **Step 2: 14x = (72)(12)**
- If we use the proportion correctly as above, the cross-product would result in \(12x = 72 \cdot 14\). Thus, this step is incorrect based on the initial proportion, which stems from the first error (A).
3. **Step 3: 14x = 864**
- This is a calculation step following from Step 2. Since Step 2 was already an error, then this could also be an error. However, it derives from the earlier incorrect setup.
4. **Step 4: x = 61.7**
- This calculates the value of \(x\) by dividing 864 by 14. But since the preceding steps were already incorrect, this value is also incorrect.
Thus, the first and fundamental error is captured in option **A**. Kara should have set up her proportion correctly based on the sides corresponding correctly based on similarity.
**Answer:** A. Kara should have written the proportion in step 1 as \(\frac{12}{72} = \frac{14}{x}\).
In similarity of triangles, the sides correspond to each other based on the established ratios. For triangles VXW and ZXY, we can set up the proportion based on the corresponding sides.
1. **Step 1: Kara writes the proportion as \(\frac{14}{72} = \frac{12}{x}\).**
- The sides WX and XY (which is 12 and \(x\)) correspond to each other. However, she should have written it as \(\frac{12}{72} = \frac{14}{x}\), because the relationship in triangle similarity is based on the corresponding sides.
- Therefore, **A** is correct: Kara should have written the proportion in step 1 as \(\frac{12}{72} = \frac{14}{x}\).
2. **Step 2: 14x = (72)(12)**
- If we use the proportion correctly as above, the cross-product would result in \(12x = 72 \cdot 14\). Thus, this step is incorrect based on the initial proportion, which stems from the first error (A).
3. **Step 3: 14x = 864**
- This is a calculation step following from Step 2. Since Step 2 was already an error, then this could also be an error. However, it derives from the earlier incorrect setup.
4. **Step 4: x = 61.7**
- This calculates the value of \(x\) by dividing 864 by 14. But since the preceding steps were already incorrect, this value is also incorrect.
Thus, the first and fundamental error is captured in option **A**. Kara should have set up her proportion correctly based on the sides corresponding correctly based on similarity.
**Answer:** A. Kara should have written the proportion in step 1 as \(\frac{12}{72} = \frac{14}{x}\).