Question

which value of x would make \\( \triangle suv \cong \triangle t uw \\) by hl?
\\( \bigcirc \\) 2
\\( \bigcirc \\) 3
\\( \bigcirc \\) 4
\\( \bigcirc \\) 5

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse - Leg) congruence states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Here, $\triangle SUV$ and $\triangle T UW$ are right triangles (right angles at $U$). The legs $SU$ and $TU$ are congruent (since $U$ is the midpoint, as indicated by the tick marks on $WV$), so we need the hypotenuses $SV$ and $WT$ to be congruent.

Step2: Set Up the Equation

The length of $SV$ is $2x + 5$ and the length of $WT$ is $4x - 1$. For $\triangle SUV \cong \triangle T UW$ by HL, we set $SV = WT$:
$$2x + 5 = 4x - 1$$

Step3: Solve for $x$

Subtract $2x$ from both sides:
$$5 = 2x - 1$$
Add 1 to both sides:
$$6 = 2x$$
Divide both sides by 2:
$$x = 3$$

Answer:

3