Question

Math

Posted 3 months ago

```
$\angle D A C=\angle B A D$.
What is the length of $\overline{A C}$ ?
Round to one decimal place.
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Given that $\angle DAC = \angle BAD$, we can conclude that $\triangle ABD$ and $\triangle ACD$ are similar by the Angle-Angle (AA) criterion for similarity

step 2

To find the length of $\overline{AC}$, we set up a proportion using corresponding sides from the similar triangles $\triangle ABD$ and $\triangle ACD$

step 3

The proportion is $\frac{AB}{AD} = \frac{AD}{AC}$. Substituting the given lengths, we get $\frac{4.9}{AD} = \frac{AD}{AC}$

step 4

We also know that $AD = BD + CD = 2.6 + 3.4 = 6.0$

step 5

Substituting $AD = 6.0$ into the proportion, we get $\frac{4.9}{6.0} = \frac{6.0}{AC}$

step 6

To solve for $AC$, cross-multiply to get $4.9 \cdot AC = 6.0 \cdot 6.0$

step 7

Simplifying, we find $AC = \frac{6.0 \cdot 6.0}{4.9}$

step 8

Calculating the value, we get $AC \approx 7.3$ when rounded to one decimal place

Answer

The length of $\overline{AC}$ is approximately 7.3 units.

Key Concept

Similar Triangles and Proportions

Explanation

The length of $\overline{AC}$ was found by using the properties of similar triangles to set up a proportion between corresponding sides and solving for the unknown length.

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