Question

if ( \triangle abccong\triangle cda ) by the hl theorem and ( ac = 84m ) and ( ad = 85m ), how long is ( overline{ba} )? (1 point) ( 27m ) ( 22m ) ( 15m ) ( 13m )

Explanation:

Step1: Apply the HL Theorem

Since \(\triangle ABC\cong\triangle CDA\) by the HL (Hypotenuse - Leg) 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 \(a=AC = 84\). Let \(CD=x\), then \(x^{2}+84^{2}=85^{2}\).

Step2: Solve for \(x\)

Expand the equation: \(x^{2}+7056 = 7225\). Subtract \(7056\) from both sides: \(x^{2}=7225 - 7056\). So \(x^{2}=169\). Take the square root of both sides: \(x=\sqrt{169}\). Since length is non - negative, \(x = 13\).

Answer:

\(13\space m\)