Question

problem 23 given: wt≅tu; ru and sw are medians of △wtu. prove: ru≅sw 1. wt≅tu; ru and sw are medians of △wtu. 1. given 2. r is the mid - point of tw and s is the mid - point of tu. 2. click here to insert 3. click here to insert 3. definition of a midpoint 4. 1/2wt = 1/2tu 4. multiplication property

Explanation:

Step1: Recall median definition

A median of a triangle is a line segment joining a vertex to the mid - point of the opposite side. So, since $\overline{RU}$ and $\overline{SW}$ are medians of $\triangle WTU$, $R$ is the mid - point of $\overline{TW}$ and $S$ is the mid - point of $\overline{TU}$.

Step2: Apply mid - point definition

By the definition of a mid - point, if $R$ is the mid - point of $\overline{TW}$, then $WR=\frac{1}{2}WT$, and if $S$ is the mid - point of $\overline{TU}$, then $TS=\frac{1}{2}TU$.

Step3: Use given equality

Given that $WT = TU$, by the multiplication property (multiplying both sides of the equation $WT = TU$ by $\frac{1}{2}$), we get $\frac{1}{2}WT=\frac{1}{2}TU$.

Step4: Substitute

Substitute the expressions from step 2 into the equation from step 3 to get $WR = TS$.

Step5: Identify isosceles triangle

Since $WT = TU$, $\triangle WTU$ is an isosceles triangle. By the Isosceles Triangle Theorem, the base angles are equal, so $\angle W=\angle T$.

Step6: Note common side

$WU$ is a common side to both $\triangle RWU$ and $\triangle STU$, so by the reflexive property, $WU = UW$.

Step7: Prove triangle congruence

We have $WR = TS$, $\angle W=\angle T$, and $WU = UW$. By the Side - Angle - Side (SAS) Congruence Postulate, $\triangle RWU\cong\triangle STU$.

Step8: Use CPCTC

Since $\triangle RWU\cong\triangle STU$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) property, $\overline{RU}\cong\overline{SW}$.

Answer:

1. Statement: $\overline{WT}\cong\overline{TU}$; $\overline{RU}$ and $\overline{SW}$ are medians of $\triangle WTU$. Reason: Given
2. Statement: $R$ is the mid - point of $\overline{TW}$ and $S$ is the mid - point of $\overline{TU}$. Reason: Definition of a median
3. Statement: $WR=\frac{1}{2}WT$ and $TS = \frac{1}{2}TU$. Reason: Definition of a mid - point
4. Statement: $\frac{1}{2}WT=\frac{1}{2}TU$. Reason: Multiplication Property (since $WT = TU$ from the given)
5. Statement: $WR=TS$. Reason: Substitution Property (from 3 and 4)
6. Statement: $\angle W=\angle T$. Reason: Isosceles Triangle Theorem (since $WT = TU$, $\triangle WTU$ is isosceles)
7. Statement: $WU = UW$ (common side). Reason: Reflexive Property
8. Statement: $\triangle RWU\cong\triangle STU$. Reason: Side - Angle - Side (SAS) Congruence Postulate ($WR = TS$, $\angle W=\angle T$, $WU = UW$)
9. Statement: $\overline{RU}\cong\overline{SW}$. Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)