Question

if \\( \triangle a b c \cong c d a \\) by the hl theorem and \\( \overline{a c}=84 \mathrm{~m} \\) and \\( \overline{a d}=85 \mathrm{~m} \\), how long is \\( \overline{b a} \\)?
(1 point)
\\( 22 \mathrm{~m} \\)
\\( 13 \mathrm{~m} \\)
\\( 27 \mathrm{~m} \\)
\\( 15 \mathrm{~m} \\)

Explanation:

Step1: Use the HL (Hypotenuse - Leg) Theorem

Since \(\triangle ABC\cong\triangle CDA\) by the HL Theorem, then \(BA = CD\). In right - triangle \(ACD\), we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = AD = 85\) (hypotenuse) and \(b=AC = 84\) (one leg). Let the other leg \(CD=x\).

Step2: Apply the Pythagorean theorem

$$x=\sqrt{AD^{2}-AC^{2}}$$

Substitute \(AD = 85\) and \(AC = 84\) into the formula:

$$x=\sqrt{85^{2}-84^{2}}=\sqrt{(85 + 84)(85 - 84)}$$

(using \(a^{2}-b^{2}=(a + b)(a - b)\))

$$x=\sqrt{(169)\times(1)}=\sqrt{169}=13$$

Answer:

\(13\) m