Question

for the triangles to be congruent by hl, what must be the value of x? 2 3 4 7

Explanation:

Step1: Recall HL congruence

HL (Hypotenuse - Leg) congruence for right - triangles means the hypotenuses and one pair of legs are equal. Let's assume the hypotenuse of the first triangle $\triangle ABC$ is $AC = 15$ and the hypotenuse of the second triangle $\triangle HFG$ is $HF=3x + 3$, and leg $BC = 12$ and leg $FG=2x + 1$. We can set up two cases: either $3x + 3=15$ (hypotenuses are equal) or $2x + 1 = 12$ (legs are equal). First, solve $3x+3 = 15$.
$3x+3=15$
Subtract 3 from both sides:
$3x=15 - 3$
$3x=12$
Divide both sides by 3:
$x = 4$
Now, check if the legs match when $x = 4$. For the second leg, when $x = 4$, $2x+1=2\times4 + 1=9$. But the other leg in the first triangle is 12. So, this is wrong.
Now solve $2x + 1=12$.
Subtract 1 from both sides:
$2x=12 - 1$
$2x=11$
$x=\frac{11}{2}$
If we set the hypotenuses equal:
$3x+3 = 15$

Step2: Solve for x

$3x=15 - 3$
$3x=12$
$x = 4$
When $x = 4$, the hypotenuse of the second triangle $3x + 3=3\times4+3=15$ (equal to the hypotenuse of the first triangle) and the leg $2x + 1=2\times4+1 = 9$ (which is the length of the other leg in the first triangle).

Answer:

4