Question

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

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse - Leg) congruence applies to right - angled triangles. For two right - angled triangles to be congruent by HL, the hypotenuse and one leg of one triangle must be equal to the hypotenuse and the corresponding leg of the other triangle.

In triangle \(ABC\) (right - angled at \(B\)) and triangle \(FGH\) (right - angled at \(G\)):

• The hypotenuse of \(\triangle ABC\) is \(AC = 15\), and one leg is \(BC=12\) (or \(AB = 9\)).
• The hypotenuse of \(\triangle FGH\) is \(FH=3x + 3\), and one leg is \(FG = 2x+1\) (or \(HG\)).

Let's assume that the hypotenuse \(AC = FH\) and the leg \(BC=FG\) (we could also match \(AB\) with \(HG\), but let's try this first).

So we set up the equations:

1. Hypotenuse: \(3x + 3=15\)
2. Leg: \(2x + 1 = 12\)

Let's solve the leg equation first: \(2x+1=12\)
Subtract 1 from both sides: \(2x=12 - 1=11\), then \(x=\frac{11}{2}=5.5\), which is not one of the options.

Now let's try matching \(AB = 9\) (leg of \(\triangle ABC\)) with \(HG\) (leg of \(\triangle FGH\)) and \(AC = FH\) (hypotenuse of \(\triangle ABC\)) with \(FH\) (hypotenuse of \(\triangle FGH\)). Wait, maybe we should match \(BC = 12\) (leg of \(\triangle ABC\)) with \(HG\) and \(AC=15\) (hypotenuse) with \(FH = 3x + 3\), and \(AB = 9\) (leg) with \(FG=2x + 1\).

Let's set \(AB=FG\), so \(2x + 1=9\)
Subtract 1 from both sides: \(2x=9 - 1 = 8\)
Divide both sides by 2: \(x = 4\)

Let's check the hypotenuse when \(x = 4\): \(3x+3=3\times4 + 3=12 + 3=15\), which is equal to the hypotenuse of \(\triangle ABC\) (\(AC = 15\)). And the leg \(FG=2x + 1=2\times4+1 = 9\), which is equal to \(AB = 9\). So the triangles will be congruent by HL when \(x = 4\).

Step2: Verify the solution

When \(x = 4\):

• \(FH=3x + 3=3\times4+3 = 15\) (equal to \(AC = 15\))
• \(FG=2x + 1=2\times4 + 1=9\) (equal to \(AB = 9\))

Since both the hypotenuse and a leg of \(\triangle FGH\) are equal to the hypotenuse and a leg of \(\triangle ABC\), the triangles are congruent by HL.

Answer:

\(x = 4\) (the option is 4)