Question

For what values of x and y are the triangles to the right congruent by HL?

Answer 1

Answer:

x = 8 and y = 4

Step-by-step explanation:

For congruency the hypotenuse and a leg must be congruent, that is

3y = x + 4 → (1) ( Equating the hypotenuse )

x = y + 4 → (2) ( equating corresponding legs )

Substitute x = y + 4 into (1)

3y = y + 4 + 4

3y = y + 8 ( subtract y from both sides )

2y = 8 ( divide both sides by 2 )

y = 4

Substitute y = 4 into (2)

x = 4 + 4 = 8

Answer 2

Based on the Hypotenuse-Length Theorem of Congruence (HL), the values of x and y that will make both right triangles congruent are:

• x = 8
• y = 4

Recall:

• Two right triangles are congruent to each other by the Hypotenuse-Length Congruence Theorem (HL) if the hypotenuse length and one leg of a right triangle is congruent to the hypotenuse length and leg of the other right triangle.

Thus, for both right triangles to be congruent, it means that their hypotenuse and one of their corresponding legs must be equal to each other.

Use this to create two equations and solve to get the value of x and y.

• Equal Hypotenuse:

$x + 4 = 3y\\\\x + 4 - 4 = 3y - 4\\\\x = 3y - 4$(eqn. 1)

• Equal legs:

$x = y + 4$ (eqn. 2).

• Find the value of y, by substituting x = (y + 4) into eqn. 1.

$x = 3y - 4$ (eqn. 1)

• Substitute

$y + 4 = 3y - 4\\\\y - 3y = -4 - 4\\\\-2y = -8\\\\y = 4$

• Find the value of x, by substituting y = 4 into eqn. 2.

$x = y + 4\\\\x = 4 + 4\\\\x = 8$

Therefore, based on the Hypotenuse-Length Theorem of Congruence (HL), the values of x and y that will make both right triangles congruent are:

• x = 8
• y = 4

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