$\angle D A C=\angle B A D$.

What is the length of $\overline{A C}$ ?
Round to one decimal place.

Question

$\angle D A C=\angle B A D$.

What is the length of $\overline{A C}$ ?
Round to one decimal place.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $\angle DAC = \angle BAD$, we can conclude that $ riangle ABD$ and $ riangle 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 $ riangle ABD$ and $ riangle 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.⚹◊What is the formula for finding the length of a side in a triangle when two angles are equal?⍭ Generate me a similar question◊

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