Question

find y, st, and tu

( y = )
( s t = )
( t u = )
(type integers or decimals.)

Explanation:

Step1: Use the property of congruent right - triangles

Since \(\angle SVT=\angle UVT\), \(ST\perp SV\), \(TU\perp UV\), by the Angle - Angle - Side (AAS) congruence criterion for right - triangles (\(\triangle STV\) and \(\triangle UTV\), \(\angle S=\angle U = 90^{\circ}\), \(\angle SVT=\angle UVT\), \(TV = TV\) (common side)), we have \(ST = TU\).

Step2: Set up the equation

Given \(ST = 4y\) and \(TU=2y + 6\). Since \(ST = TU\), we set up the equation \(4y=2y + 6\).
Subtract \(2y\) from both sides: \(4y-2y=2y + 6-2y\).
We get \(2y=6\).
Divide both sides by 2: \(y=\frac{6}{2}=3\).

Step3: Calculate \(ST\) and \(TU\)

Substitute \(y = 3\) into the expressions for \(ST\) and \(TU\).
For \(ST\): \(ST = 4y\), so \(ST=4\times3 = 12\).
For \(TU\): \(TU=2y + 6\), so \(TU=2\times3+6=6 + 6=12\).

Answer:

\(y = 3\), \(ST = 12\), \(TU = 12\)