Question

what values of x and y make δstu ≅ δfhg?
triangle stu with right angle at u, side tu labeled x + 48, side ts labeled 2y + x + 23.
triangle fhg with right angle at g, side hg labeled 5x + 28, side fh labeled 10x + y - 7.
find x and y.

Explanation:

Step1: Identify Corresponding Sides

Since \(\triangle STU \cong \triangle FHG\), their corresponding sides are equal. In right triangles, the legs and hypotenuse correspond. So, \(TU = HG\) and \(TS = FH\), \(SU = FG\) (but \(SU\) and \(FG\) are legs, \(TU\) and \(HG\) are legs, \(TS\) and \(FH\) are hypotenuses). So:

• \(TU = HG\): \(x + 48 = 5x + 28\)
• \(TS = FH\): \(2y + x + 23 = 10x + y - 7\)

Step2: Solve for \(x\) from \(x + 48 = 5x + 28\)

Subtract \(x\) from both sides: \(48 = 4x + 28\)
Subtract 28: \(20 = 4x\)
Divide by 4: \(x = 5\)

Step3: Substitute \(x = 5\) into \(2y + x + 23 = 10x + y - 7\)

Substitute \(x = 5\): \(2y + 5 + 23 = 10(5) + y - 7\)
Simplify: \(2y + 28 = 50 + y - 7\)
Simplify right side: \(2y + 28 = 43 + y\)
Subtract \(y\): \(y + 28 = 43\)
Subtract 28: \(y = 15\)

Answer:

\(x = 5\), \(y = 15\)