Question

1. find the values of x and y that make these triangles congruent by the hl theorem. x = 3, y = 6 x = 6, y = 3 x = 2, y = 1 x = 3, y = 2

Explanation:

Step1: Set up equations based on HL Theorem

For right - angled triangles, by the HL (Hypotenuse - Leg) Theorem, the hypotenuses and one pair of legs are equal.
We get the system of equations:

$$\begin{cases} 3y + 1=x + 4\\ 2y=x \end{cases}$$

Step2: Substitute \(x = 2y\) into the first equation

Substitute \(x = 2y\) into \(3y + 1=x + 4\).
We have \(3y+1=(2y)+4\).
Simplify the equation: \(3y+1 = 2y+4\).
Subtract \(2y\) from both sides: \(3y-2y+1=2y - 2y+4\), so \(y+1 = 4\).
Subtract 1 from both sides: \(y=4 - 1=3\).

Step3: Find the value of \(x\)

Since \(x = 2y\) and \(y = 3\), then \(x=2\times3 = 6\).

Answer:

\(x = 6,y = 3\)