Question

what values of b and c make $\triangle hij \cong \triangle zyx$?

(image of two right triangles: $\triangle hij$ with right angle at i, side ij labeled $c + 15$, side hj labeled $9b + 37$; $\triangle zyx$ with right angle at y, side xy labeled $19c - 3$, side zx labeled $b + 45$)

$b = \square$
$c = \square$

Explanation:

Step1: Set up equation for \( c \)

Since \( \triangle HIJ \cong \triangle ZYX \), their corresponding sides are equal. The sides \( IJ \) (length \( c + 15 \)) and \( YX \) (length \( 19c - 3 \)) are corresponding. So we set up the equation:
\( c + 15 = 19c - 3 \)
Subtract \( c \) from both sides:
\( 15 = 18c - 3 \)
Add 3 to both sides:
\( 18 = 18c \)
Divide both sides by 18:
\( c = 1 \)

Step2: Set up equation for \( b \)

The sides \( HJ \) (length \( 9b + 37 \)) and \( ZX \) (length \( b + 45 \)) are corresponding. So we set up the equation:
\( 9b + 37 = b + 45 \)
Subtract \( b \) from both sides:
\( 8b + 37 = 45 \)
Subtract 37 from both sides:
\( 8b = 8 \)
Divide both sides by 8:
\( b = 1 \)

Answer:

\( b = 1 \)
\( c = 1 \)