Question

Please answer soon. Include Statement and Reason if possible.

Answer 1

Answer:  AC = sqrt(21)/2

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Explanation:

Triangle CDA is congruent to triangle CDB. We can use the HL (hypotenuse leg) congruence theorem to prove this. This only works because we have two right triangles.

Since CDA and CDB are congruent, this means their corresponding pieces are the same length. Specifically AD = DB, so

AD+DB = AB

AD+AD = 3

2*AD = 3

AD = 3/2 = 1.5

For triangle CDA, we have AD = 3/2 = 1.5 and CD = sqrt(3). We can use the pythagorean theorem to find the missing side AC

a^2 + b^2 = c^2

(AD)^2 + (CD)^2 = (AC)^2

(3/2)^2 + (sqrt(3))^2 = (AC)^2

9/4 + 3 = (AC)^2

(AC)^2 = 9/4 + 3

(AC)^2 = 9/4 + 12/4

(AC)^2 = 21/4

AC = sqrt(21/4)

AC = sqrt(21)/sqrt(4)

AC = sqrt(21)/2

This is the same as writing (1/2)*sqrt(21) or 0.5*sqrt(21)