Question

if △abc ≅ △cda by hl theorem and $overline{ac}=21 m$ and $overline{ad}=29 m$, how long is $overline{ba}$ (1 point) 36 m 18 m 20 m 24 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, $AC$ is a common hypotenuse for both right - triangles $\triangle ABC$ and $\triangle CDA$, and $BA$ and $CD$ are legs, $AD$ and $BC$ are legs.

Step2: Apply Pythagorean theorem in $\triangle ABC$

In right - triangle $\triangle ABC$, by the Pythagorean theorem $BA=\sqrt{BC^{2}-AC^{2}}$. Since $\triangle ABC\cong\triangle CDA$, we can also consider right - triangle $\triangle CDA$. In right - triangle $\triangle CDA$, if the hypotenuse $AD = 29m$ and one leg $AC=21m$, and we want to find the other leg (which is congruent to $BA$). According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a$ and $b$ are legs. Let the unknown leg be $x$, then $x=\sqrt{AD^{2}-AC^{2}}$.

Step3: Calculate the length of $BA$

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

Answer:

$20m$