Question

independent work

1. the following two triangles are congruent through hl.

find the value of x, then find the length of the hypotenuse.
2.) congruent polygons wbh ym and tchrk are shown where \\( \overline { t c } \cong \overline { t k } \\).
determine the length of \\( \overline { t c } \\) if \\( t c = 8 x - 11 \\) and \\( t k = 3 x + 24 \\).

Explanation:

Step1: Solve for \( x \) in the first problem

Since the two right - triangles are congruent by HL (Hypotenuse - Leg), their corresponding legs are equal. So, \( 3x + 4=6x - 47 \).
Subtract \( 3x \) from both sides: \( 4 = 3x-47 \).
Add \( 47 \) to both sides: \( 3x=4 + 47=51 \).
Divide both sides by \( 3 \): \( x = 17 \).

Step2: Find the length of the hypotenuse

Substitute \( x = 17 \) into \( 3x + 4 \) (or \( 6x - 47 \)). Using \( 3x+4 \), we have \( 3\times17 + 4=51 + 4=55 \).

Step3: Solve for \( x \) in the second problem

Since \( \overline{TC}\cong\overline{TK} \) (given \( TC = 8x-11 \) and \( TK = 3x + 24 \)), then \( 8x-11=3x + 24 \).
Subtract \( 3x \) from both sides: \( 5x-11=24 \).
Add \( 11 \) to both sides: \( 5x=24 + 11=35 \).
Divide both sides by \( 5 \): \( x = 7 \).

Step4: Find the length of \( \overline{TC} \)

Substitute \( x = 7 \) into \( TC = 8x-11 \). So, \( TC=8\times7-11=56 - 11=45 \).

Answer:

1. \( x = 17 \), length of the hypotenuse \(=55\).
2. Length of \( \overline{TC}=45 \).