Question

Find the value of m and n that prove the two triangles are congruent by the HL theorem.

Answer 1

If both triangles are congruent by the HL theorem, then their hypotenuses are equal and at least one of the corresponding legs is equal too.

Hypothenuses:

`13=4m+1`

From this expression, you can calculate the value of m

`\begin{gathered} 13=4m+1 \\ 13-1=4m \\ 12=4m \\ (12)/(4)=(4m)/(4) \\ 3=m \end{gathered}`

Legs:

`2m+n=8m-2n`

Replace the expression with the calculated value of m

`\begin{gathered} 2\cdot3+n=8\cdot3-2n \\ 6+n=24-2n \end{gathered}`

Now pass the n-related term to the left side of the equation and the numbers to the right side:

`\begin{gathered} 6-6+n=24-6-2n \\ n=18-2n \\ n+2n=18-2n+2n \\ 3n=18 \end{gathered}`

And divide both sides of the expression by 3

`\begin{gathered} (3n)/(3)=(18)/(3) \\ n=6 \end{gathered}`

So, for m=3 and n=6 the triangles are congruent by HL