Question

for what values of x and y are the triangles to the right congruent by hl?
x = \square and y = \square

Explanation:

Step1: Set up equations for HL congruence

For right - triangles to be congruent by HL (Hypotenuse - Leg), the hypotenuses and one pair of legs must be equal.
Set \(x = 4y\) (legs) and \(x + 2=y + 1\) (hypotenuses).

Step2: Substitute \(x = 4y\) into \(x + 2=y + 1\)

Substitute \(x\) in the second equation: \(4y+2=y + 1\).
Subtract \(y\) from both sides: \(4y - y+2=y - y + 1\), so \(3y+2 = 1\).
Subtract 2 from both sides: \(3y+2 - 2=1 - 2\), then \(3y=-1\). (This is wrong, we should use the correct correspondence. The correct correspondence is \(x=y + 1\) (leg) and \(x + 2=4y\) (hypotenuse))
Correct step:
Substitute \(x=y + 1\) into \(x + 2=4y\).
We get \((y + 1)+2=4y\).
Simplify: \(y+3 = 4y\).
Subtract \(y\) from both sides: \(y - y+3=4y - y\), so \(3y=3\).
Divide both sides by 3: \(y = 1\).

Step3: Find \(x\)

Since \(x=y + 1\) and \(y = 1\), then \(x=1 + 1=2\).

Answer:

\(x = 2\) and \(y = 1\)