Here are the step-by-step solutions for each problem on the worksheet. These problems use the
Triangle Midsegment Theorem, which says that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is
half the length of that third side.
1. Find $PQ$
*
Identify the parts: In triangle $GHI$, points $P$ and $Q$ are midpoints (indicated by the tick marks). This makes segment $PQ$ a midsegment.
*
Apply the rule: The midsegment ($PQ$) is half the length of the base ($GI$).
*
Calculate: The base $GI$ is 18.
$$PQ = \frac{18}{2} = 9$$
*
Answer: $PQ = 9$ (yellow)
2. Find $UV$
*
Identify the parts: In triangle $RST$, $U$ and $V$ are midpoints. Segment $UV$ is the midsegment.
*
Apply the rule: The midsegment ($UV$) is half the length of the base ($RT$).
*
Calculate: The base $RT$ is 12.
$$UV = \frac{12}{2} = 6$$
*
Answer: $UV = 6$ (blue)
3. Find $SU$
*
Identify the parts: In triangle $STU$, $W$ and $X$ are midpoints. Segment $WX$ is the midsegment with length 7. We need to find the base $SU$.
*
Apply the rule: The midsegment is half the base, which means the base is
double the midsegment.
*
Calculate:
$$SU = 7 \times 2 = 14$$
*
Answer: $SU = 14$ (black)
4. Find $IK$
*
Identify the parts: In triangle $IJK$, $X$ and $W$ are midpoints. Segment $XW$ is the midsegment with length 5. We need to find the base $IK$.
*
Apply the rule: The base is double the midsegment.
*
Calculate:
$$IK = 5 \times 2 = 10$$
*
Answer: $IK = 10$ (orange)
5. Find $EF$
*
Identify the parts: In triangle $RST$, $E$ and $F$ are midpoints. Segment $EF$ is the midsegment. The base is side $ST$.
*
Find the base length: The diagram shows the length of side $ST$ is 30. (Note: The numbers 7 and 32 refer to other sides and are not needed for this specific calculation).
*
Apply the rule: The midsegment ($EF$) is half the base ($ST$).
*
Calculate:
$$EF = \frac{30}{2} = 15$$
*
Answer: $EF = 15$ (pink)
6. Find $x$
*
Identify the parts: In triangle $CDE$, $H$ and $G$ are midpoints. Segment $HG$ is the midsegment. The base is side $CE$.
*
Set up the equation: The midsegment ($HG$) is half the length of the base ($CE$). Alternatively, the base is twice the midsegment.
* Base $CE = x + 3$
* Midsegment $HG = x - 1$
* Equation: $x + 3 = 2(x - 1)$
*
Solve for $x$:
1. Distribute the 2 on the right side:
$$x + 3 = 2x - 2$$
2. Subtract $x$ from both sides:
$$3 = x - 2$$
3. Add 2 to both sides:
$$5 = x$$
*
Check: If $x = 5$, then Base $= 5 + 3 = 8$ and Midsegment $= 5 - 1 = 4$. Since 4 is half of 8, the answer is correct.
*
Answer: $x = 5$ (purple)
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Final Answer:
1. PQ = 9
2. UV = 6
3. SU = 14
4. IK = 10
5. EF = 15
6. x = 5
Parent Tip: Review the logic above to help your child master the concept of midsegment of a triangle worksheets.
