Math  /  Geometry

QuestionAnalyzing a Solution
Medea found the missing measure using a proportion of the reduced trapezoid.
1. Proportion: 4x=812\frac{4}{x}=\frac{8}{\frac{1}{2}}
2. Cross products: 4(12)=8(x)4\left(\frac{1}{2}\right)=8(x)
3. Simplify: 2=8x2=8 x
4. Divide: 14=x\frac{1}{4}=x

Analyze Medea's work. In which step did Medea make an error? In step 1, the corresponding parts are in different positions. In step 2, the cross products should be multiplied instead of being equal. In step 3, she should have added instead of multiplying to simplify. In step 4, she needed to divide 2 to both sides to get x=4x=4.

Studdy Solution

STEP 1

What is this asking? We need to find the mistake Medea made when solving her proportion. Watch out! Proportions can be tricky!
Make sure the corresponding sides match up correctly.

STEP 2

1. Analyze the proportion setup.
2. Examine the cross-product.
3. Check the simplification.
4. Verify the final solution.

STEP 3

Medea set up the proportion: 4x=812\frac{4}{x} = \frac{8}{\frac{1}{2}}.
Let's think about what these numbers represent.
It looks like she's comparing corresponding sides of two similar trapezoids.
We need to make sure she matched up the right sides!
The **numerator** of the left side, 44, should correspond to the **numerator** on the right side, 88, and the **denominator** on the left, xx, should correspond to the **denominator** on the right, 12\frac{1}{2}.
Does this make sense given the problem?
It seems like 88 is the longer side and 44 is the shorter side, so 12\frac{1}{2} should also be the shorter side, and xx should be the longer side.
Uh oh, it looks like Medea flipped the relationship!

STEP 4

Medea wrote 412=8x4 \cdot \frac{1}{2} = 8 \cdot x.
This is the correct way to do the cross-product.
If we have ab=cd\frac{a}{b} = \frac{c}{d}, then ad=bca \cdot d = b \cdot c.
So, even though Medea's initial proportion was wrong, her cross-product is mathematically correct *given* her incorrect proportion.

STEP 5

Medea simplified 412=8x4 \cdot \frac{1}{2} = 8 \cdot x to 2=8x2 = 8x.
This is perfectly fine!
Four times one-half is indeed two.

STEP 6

Medea divided both sides of 2=8x2 = 8x by 88 to get 28=x\frac{2}{8} = x, which simplifies to 14=x\frac{1}{4} = x.
This step is also correct *given* the previous steps.

STEP 7

Medea made her mistake in the **very first step**, setting up the proportion.
She flipped the relationship between the corresponding sides.
The correct proportion should have been 4x=128\frac{4}{x} = \frac{\frac{1}{2}}{8}.

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