Question

for the triangles to be congruent by hl, what must be the value of x?
○ 2
○ 3
○ 4
○ 7
(diagram: two right triangles, first with right angle at b, legs 9 (ab), 12 (bc), hypotenuse 15 (ac); second with right angle at g, leg fg = 2x + 1, hypotenuse fh = 3x + 3)

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse-Leg) congruence for right triangles requires hypotenuse and one leg to be equal. In $\triangle ABC$ (right at $B$) and $\triangle FGH$ (right at $G$), hypotenuse $AC = 15$, leg $AB = 9$, $BC = 12$; hypotenuse $FH = 3x + 3$, leg $FG = 2x + 1$, $HG$ (corresponding to $AB$). So either hypotenuse and one leg must match. Let's match hypotenuse $FH = AC = 15$ and leg $FG = BC = 12$ (or $HG = AB = 9$). Let's try $FG = 12$: $2x + 1 = 12$ → $2x = 11$ → $x = 5.5$ (not an option). Try $FH = 15$ and $HG = 9$ (but $HG$ is vertical, maybe $FG = BC = 12$? Wait, maybe $FH$ (hypotenuse) should equal $AC = 15$, and $FG$ (leg) equal $BC = 12$? Wait, no, let's check the legs. Wait, $\triangle ABC$: legs $AB = 9$, $BC = 12$; hypotenuse $AC = 15$. $\triangle FGH$: legs $FG = 2x + 1$, $HG$; hypotenuse $FH = 3x + 3$. For HL, either:

1. Hypotenuse $FH = AC = 15$ and leg $FG = BC = 12$ (so $3x + 3 = 15$ and $2x + 1 = 12$? Wait, no, HL is hypotenuse and one leg. So either hypotenuse and one leg. Let's solve $3x + 3 = 15$ (hypotenuse) and check leg.

Step2: Solve for hypotenuse

If $FH = AC = 15$, then $3x + 3 = 15$. Subtract 3: $3x = 12$. Divide by 3: $x = 4$. Now check leg $FG$: $2x + 1 = 2(4) + 1 = 9$. Wait, $\triangle ABC$ has leg $AB = 9$, so $FG = 9$ (matching $AB$) and hypotenuse $FH = 15$ (matching $AC$). So that works. Let's verify: $x = 4$, $FH = 3(4)+3 = 15$, $FG = 2(4)+1 = 9$. So $\triangle FGH$ has hypotenuse 15, leg 9, right triangle. $\triangle ABC$ has hypotenuse 15, leg 9 (AB), so by HL, they are congruent.

Answer:

4 (corresponding to the option "4")