Question

if (mangle{sut}=mangle{suv}=70^{circ}, st = w + 56), and (sv = 3w), what is (sv)?

Explanation:

Step1: Identify congruent triangles

Since $\angle SUT=\angle SUV = 70^{\circ}$ and $\angle SVU=\angle STU = 90^{\circ}$ and $SU$ is common to both $\triangle SUV$ and $\triangle SUT$, by the AAS (Angle - Angle - Side) congruence criterion, $\triangle SUV\cong\triangle SUT$.

Step2: Set equal corresponding sides

Corresponding sides of congruent triangles are equal. So $ST = SV$.
We know that $ST=w + 56$ and $SV = 3w$. Then we set up the equation $w + 56=3w$.

Step3: Solve the equation for $w$

Subtract $w$ from both sides: $56=3w - w$, which simplifies to $56 = 2w$.
Divide both sides by 2: $w=\frac{56}{2}=28$.

Step4: Find the length of $SV$

Substitute $w = 28$ into the expression for $SV$. Since $SV = 3w$, then $SV=3\times28 = 84$.

Answer:

84