Answer:
The length of AC = 18 meters.
Step-by-step explanation:
Given : ∠ABC ≅ ∠ACD, BC = 12 m and AD = 27 m
To find : AC
Solution : Since, the bases of the trapezoid are parallel to each other
⇒ AD ║ BC
So, ∠ACB = ∠CAD ( Alternate interior angles are equal)
Now, in ΔACB and ΔDAC,
∠ABC ≅ ∠ACD (Given)
∠ACB = ∠DAC (Proved above)
So, BY AA postulate of similarity of triangles, ΔACB ~ ΔDAC
Since, the sides of similar triangles are proportion to each other
[tex]\implies \frac{AC}{DA}=\frac{CB}{AC}\\\\\implies AC^2=AD\times BC\\\\\implies AC^2=27\times 12\\\implies AC^2=324\\\bf\implies AC=18m[/tex]
Hence, the length of AC = 18 meters.
Since ΔABC and ΔACD are similar triangles, therefore the length of AC is: 18 m.
Similar triangles are triangles with corresponding sides that have the same ratio.
Thus:
AD is parallel to BC (bases of trapezoid are parallel)
Therefore:
∠ACB = ∠CAD (alternate interior angles)
This implies that, ΔABC ~ ΔACD by AA similarity theorem.
Thus:
AC/DA = CB/AC
Substitute
AC² = 12 × 27
AC = √324
AC = 18 m
Therefore, since ΔABC and ΔACD are similar triangles, therefore the length of AC is: 18 m.
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