Question

if △abc≅△cda by hl theorem and $overline{ac}=21m$ and $overline{ad}=29m$, how long is $overline{ba}$ (1 point) 36 m 18 m 24 m 20 m

Explanation:

Step1: Recall HL - Hypotenuse - Leg theorem

In right - triangles $\triangle ABC$ and $\triangle CDA$, if $\triangle ABC\cong\triangle CDA$ by HL theorem, the hypotenuse and one leg of one right - triangle are congruent to the hypotenuse and one leg of the other right - triangle. Here, $AD$ is the hypotenuse of $\triangle CDA$ and $AD$ is also the hypotenuse of $\triangle ABC$. $AC$ is a common leg for both right - triangles.

Step2: Apply congruence property

Since $\triangle ABC\cong\triangle CDA$ by HL theorem, the corresponding legs are equal. In right - triangle $\triangle ABC$, if $AC$ is one leg and $BA$ is the other leg, and in right - triangle $\triangle CDA$, $AC$ is one leg and $CD$ is the other leg. Also, since $\triangle ABC\cong\triangle CDA$, we can use the Pythagorean theorem in $\triangle ABC$. Let $BA = x$. In right - triangle $\triangle ABC$, by the Pythagorean theorem $AB^{2}+AC^{2}=BC^{2}$. But since $\triangle ABC\cong\triangle CDA$, we know that if the hypotenuse $AD = 29m$ and one leg $AC = 21m$, then using the Pythagorean theorem $AB=\sqrt{AD^{2}-AC^{2}}$.

Step3: Calculate the length of $BA$

Substitute $AD = 29$ and $AC = 21$ into the formula $AB=\sqrt{AD^{2}-AC^{2}}=\sqrt{29^{2}-21^{2}}=\sqrt{(29 + 21)(29 - 21)}=\sqrt{50\times8}=\sqrt{400}=20m$.

Answer:

20 m