Use the Hypotenuse-Leg Theorem to solve for the values of s and t that make the triangles congruent. The first hypotenuse BD has a length of -5s + 27t. The first leg CD has a length of -2s – 2t + 94 and the second leg QR has a length of -1t + s + 14. For the triangles to be congruent, the hypotenuse lengths must be equal and the leg lengths must be equal. Set BD equal to PR and CD equal to QR, and solve this system of equations for s and t.
t + s + 64 = -5s + 27t
-2s – 2t + 94 = -1t + s + 14
Since each equation has s and t terms on both sides, start by rewriting the system of equations in standard form. First, rewrite the hypotenuse equation in standard form.
To solve for the values of ( s ) and ( t ) using the Hypotenuse-Leg Theorem, we first need to set up the equations based on the information provided and rewrite them in standard form.
Given:
1. Hypotenuse equation: ( -5s + 27t = s + t + 64 )
2. Leg equation: ( -2s – 2t + 94 = -1t + s + 14 )
### Step 1: Rewrite the hypotenuse equation
Starting with the hypotenuse equation:
[ -5s + 27t – s – t – 64 = 0 ]
Combine like terms:
[ -6s + 26t – 64 = 0 ]
To get the standard form:
[ 6s – 26t + 64 = 0 ]
### Step 2: Rewrite the leg equation
Now for the leg equation:
[ -2s – 2t + 94 = -1t + s + 14 ]
Rearranging gives:
[ -2s – 2t + 94 + 1t – s – 14 = 0 ]
Combine like terms:
[ -3s – t + 80 = 0 ]
Rewrite in standard form:
[ 3s + t – 80 = 0 ]
### Step 3: System of equations
We now