Question

which value of x would make △suv ≅ △tuw by hl? 3 4 2 5

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse-Leg) congruence for right triangles: if hypotenuse and one leg of a right triangle are equal to hypotenuse and one leg of another right triangle, they are congruent. Here, \( \triangle SUV \) and \( \triangle TUW \) are right triangles. The legs \( SU \) and \( TU \) are equal (marked with ticks), so hypotenuses \( SV \) and \( WT \) must be equal.

Step2: Set Up Equation

Set \( SV = WT \): \( 2x + 9 = 4x - 1 \).

Step3: Solve for \( x \)

Subtract \( 2x \) from both sides: \( 9 = 2x - 1 \). Add 1 to both sides: \( 10 = 2x \). Divide by 2: \( x = 5 \). Wait, but let's check again. Wait, maybe I mixed up? Wait, no—wait, the hypotenuses: \( SV = 2x + 9 \), \( WT = 4x - 1 \). Wait, but maybe the legs? Wait, no, the right angles are at U. So \( \triangle SUV \) has legs \( SU \) and \( UV \), hypotenuse \( SV \). \( \triangle TUW \) has legs \( TU \) and \( UW \), hypotenuse \( WT \). Since \( SU = TU \) (vertical angles? Wait, no, the ticks: \( UW = UV \) (marked with ticks). So legs: \( UV = UW \), and right angles. So hypotenuses \( SV = WT \). So equation \( 2x + 9 = 4x - 1 \). Solving: \( 2x = 10 \), \( x = 5 \). Wait, but the options include 5? Wait, the last option is 5? Let me check the options again. The options are 3,4,2,5. So \( x = 5 \). Wait, but let's verify: if \( x = 5 \), \( SV = 2(5)+9 = 19 \), \( WT = 4(5)-1 = 19 \). Yes, equal. So that works.

Answer:

5 (assuming the last option is 5, e.g., D. 5 if options are A.3, B.4, C.2, D.5)