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Step-by-step solution for: Medians and Centroid Notes
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Show Answer Key & Explanations
Step-by-step solution for: Medians and Centroid Notes
To find the missing measures, we need to use the properties of the centroid of a triangle.
Key Rule:
The centroid is the point where the three medians intersect. A median connects a vertex to the midpoint of the opposite side. The centroid divides each median into two parts with a specific ratio:
1. The part from the vertex to the centroid is 2/3 of the total length of the median.
2. The part from the centroid to the midpoint is 1/3 of the total length of the median.
3. Therefore, the longer part (vertex to centroid) is exactly twice the length of the shorter part (centroid to midpoint).
Let's solve for each missing measure in Problem 2 step-by-step.
Given:
* $H$ is the centroid of $\triangle TUV$.
* $TZ = 60$ ($Z$ is on side $TU$, so $VZ$ is the median from vertex $V$). Wait, looking at the diagram, $Z$ is on side $TU$. So the median is segment $VZ$. The problem states $TZ=60$. Since $VZ$ is a median, $Z$ is the midpoint of $TU$. This means $TZ = ZU$.
* $XZ = 28$ ($X$ is on side $TV$? No, let's look closer. $X$ is on side $TV$. The segment connecting $U$ to $X$ passes through $H$. So $UX$ is a median. $H$ lies on $UX$. The problem gives $XZ=28$? That seems odd. Let's re-read carefully. Ah, the text says "$XZ = 28$". Looking at the diagram, $X$ is on side $TV$ and $Z$ is on side $TU$. $XZ$ is a segment connecting two midpoints? No, usually these problems give segments along the medians. Let's look at the labels again.
* Median from $V$ goes to $Z$ on $TU$. So $V-H-Z$ is a line.
* Median from $T$ goes to $Y$ on $UV$. So $T-H-Y$ is a line.
* Median from $U$ goes to $X$ on $TV$. So $U-H-X$ is a line.
* The text says: $TZ = 60$, $XZ = 28$, and $WZ = 25$. Wait, there is no $W$ in the main triangle vertices. Let's look at the diagram for Problem 2 again. There is a point $W$ on the side $TV$? No, $W$ is on the side $TV$ in the diagram? Actually, looking at the diagram for #2:
* Vertices are $T, U, V$.
* Midpoints appear to be $X$ (on $TV$), $Y$ (on $UV$), and $Z$ (on $TU$).
* Medians are $UZ$, $TY$, and $VX$? Or $VZ$? Let's trace the lines.
* Line from $V$ goes to $Z$ on $TU$. So $VZ$ is a median.
* Line from $T$ goes to... it looks like it goes to a point on $UV$. Let's call it $Y$. So $TY$ is a median.
* Line from $U$ goes to... it looks like it goes to a point on $TV$. Let's call it $X$? But the text says $XZ=28$. If $X$ and $Z$ are midpoints, $XZ$ is a midsegment. But the question asks for lengths like $ZV$, $ZY$, etc.
* Let's re-read the text provided in the image for #2: "If $H$ is the centroid of $\triangle TUV$, $TZ = 60$, $XZ = 28$, and $WZ = 25$..."
* This text is confusing because $W$ is not a standard label. Let's look at the diagram labels very closely.
* In diagram 2:
* Vertex $T$ top left.
* Vertex $U$ bottom right? No, $U$ is top right.
* Vertex $V$ bottom.
* Point $Z$ is on side $TU$.
* Point $X$ is on side $TV$? Or is $X$ on $TU$?
* Let's look at the lines intersecting at $H$.
* One line comes from $V$, goes through $H$, to $Z$ on $TU$. So $VZ$ is a median.
* One line comes from $T$, goes through $H$, to a point on $UV$. Let's assume this point is $Y$. So $TY$ is a median.
* One line comes from $U$, goes through $H$, to a point on $TV$. Let's assume this point is $W$? Or $X$?
* The text lists $XZ=28$ and $WZ=25$. This implies $X$ and $W$ are points related to $Z$.
* Actually, let's look at the previous problem (#1) to understand the notation style. In #1, $P$ is centroid. $JK=22$ (side), $KN=13$ (part of median?), $OL=18$ (part of median?).
* Let's re-examine Problem 2's text and diagram together.
* Diagram 2 shows medians intersecting at $H$.
* The vertices are $T, U, V$.
* The points on the sides are $X, Y, Z$.
* Usually, $X$ is opposite $T$, $Y$ opposite $U$, $Z$ opposite $V$. But here:
* $Z$ is on side $TU$. So the median is from $V$ to $Z$. Thus $VZ$ is a median.
* The line from $T$ goes to side $UV$. Let's call the intersection $Y$. So $TY$ is a median.
* The line from $U$ goes to side $TV$. Let's call the intersection $X$? Or $W$?
* The text says: "$TZ = 60, XZ = 28, \text{and } WZ = 25$".
* This is extremely likely a typo in the book or my reading. Let's look at the letters again.
* Maybe the points on the sides are $W, X, Y$?
* Let's look at the diagram labels again.
* Side $TU$ has point $Z$.
* Side $UV$ has point $Y$? (Hard to see, looks like a letter near the midpoint of UV).
* Side $TV$ has point $X$? Or $W$?
* Let's assume the standard convention:
* Median from $V$ hits $TU$ at $Z$. So $Z$ is midpoint of $TU$.
* Median from $T$ hits $UV$ at $Y$. So $Y$ is midpoint of $UV$.
* Median from $U$ hits $TV$ at $X$ (or $W$). Let's assume the point on $TV$ is $W$ based on the text "$WZ=25$" being weird if $W$ is on $TV$.
* Wait, look at the text again: "$TZ = 60, XZ = 28, \text{and } WZ = 25$".
* Could $X$ and $W$ be points on the medians?
* Let's look at the questions asked:
a) $ZV =$ ?
b) $ZY =$ ?
c) $ZU =$ ?
d) $TY =$ ?
e) $XY =$ ?
* This suggests $X, Y, Z$ are the midpoints of the sides.
* If $Z$ is midpoint of $TU$, then $TZ = ZU$.
* If $Y$ is midpoint of $UV$, then $UY = YV$.
* If $X$ is midpoint of $TV$, then $TX = XV$.
* Let's re-read the given values with this hypothesis:
* "$TZ = 60$": Since $Z$ is midpoint of $TU$, $ZU = 60$ and total side $TU = 120$.
* "$XZ = 28$": $X$ and $Z$ are midpoints of sides $TV$ and $TU$. The segment connecting midpoints of two sides is parallel to the third side ($UV$) and half its length. So $XZ = \frac{1}{2} UV$. This would mean $UV = 56$. And $Y$ is midpoint of $UV$, so $UY = YV = 28$. Also $ZY$ connects midpoints of $TU$ and $UV$, so $ZY = \frac{1}{2} TV = TX = XV$. And $XY$ connects midpoints of $TV$ and $UV$, so $XY = \frac{1}{2} TU = TZ = ZU = 60$.
* "$WZ = 25$": Who is $W$? Looking at the diagram, there is a point labeled $W$ on the median from $U$? Or is $W$ the point on side $TV$? If $W$ is the point on side $TV$, then $W$ is the midpoint. So $W=X$? Why two names?
* Let's look really closely at the image text for #2. It says: "If $H$ is the centroid of $\triangle TUV$, $TZ = 60$, $XZ = 28$, and $WZ = 25$..."
* Wait, is it possible the text says $HZ = 25$? The letter 'W' and 'H' can look similar in some fonts or handwriting, but this is printed text. However, $H$ is the centroid. $Z$ is on the side. $HZ$ is the segment from centroid to side.
* Let's test the hypothesis: Given $TZ=60$, $XZ=28$, and $HZ=25$.
* If $HZ = 25$: $HZ$ is the shorter part of median $VZ$. The longer part $VH$ is $2 \times HZ = 50$. Total median $VZ = VH + HZ = 75$.
* Question (a) asks for $ZV$. $ZV$ is the whole median? Or just the segment from Z to V? Yes, $ZV$ is the median length. So $ZV = 75$.
* Question (c) asks for $ZU$. Since $VZ$ is a median, $Z$ is the midpoint of $TU$. Given $TZ = 60$, then $ZU = 60$.
* Question (b) asks for $ZY$. $Z$ is midpoint of $TU$. $Y$ is likely midpoint of $UV$. $ZY$ is a midsegment. Length $ZY = \frac{1}{2} TV$. We don't know $TV$ yet.
* What about $XZ = 28$? If $X$ is midpoint of $TV$ and $Z$ is midpoint of $TU$, then $XZ$ is a midsegment parallel to $UV$. $XZ = \frac{1}{2} UV$. So $UV = 56$. Since $Y$ is midpoint of $UV$, $UY = YV = 28$.
* Question (e) asks for $XY$. $X$ is midpoint of $TV$, $Y$ is midpoint of $UV$. $XY$ is midsegment parallel to $TU$. $XY = \frac{1}{2} TU$. Since $TU = TZ + ZU = 60 + 60 = 120$, then $XY = 60$.
* Question (d) asks for $TY$. $TY$ is the median from $T$ to side $UV$. We need the length of median $TY$. Do we have enough info? We know sides $TU=120$, $UV=56$. We don't know $TV$.
* Is there another interpretation?
* Let's look at the "W" again. In the diagram, the point on the side $TV$ is labeled $W$? Or $X$?
* Let's look at the letters in the diagram for #2.
* Top-left vertex: $T$.
* Top-right vertex: $U$.
* Bottom vertex: $V$.
* Point on $TU$: $Z$.
* Point on $UV$: $Y$? (Looks like a Y).
* Point on $TV$: $W$? (Looks like a W).
* So the midpoints are $Z, Y, W$.
* Then what is $X$? The text mentions $XZ=28$. And question (e) asks for $XY$. This implies $X$ is a point.
* Maybe the point on $TV$ is $X$? And the text has a typo saying $WZ$?
* Or maybe the point on $TV$ is $W$, and the text has a typo saying $XZ$?
* Let's look at the questions again: a) $ZV$, b) $ZY$, c) $ZU$, d) $TY$, e) $XY$.
* The questions use $X$ and $Y$. This strongly suggests the midpoints are named $X, Y, Z$.
* So, $X$ is on $TV$, $Y$ is on $UV$, $Z$ is on $TU$.
* This makes $W$ in the text "$WZ=25$" the outlier. It is highly probable that $W$ is a typo for $H$. $H$ is the centroid. $HZ$ is the distance from centroid to midpoint $Z$.
* Let's proceed with: Midpoints are $X$ (on $TV$), $Y$ (on $UV$), $Z$ (on $TU$). Centroid is $H$.
* Given:
1. $TZ = 60$. Since $Z$ is midpoint of $TU$, $ZU = 60$ and $TU = 120$.
2. $XZ = 28$. Segment connecting midpoints $X$ (of $TV$) and $Z$ (of $TU$). By Midsegment Theorem, $XZ = \frac{1}{2} UV$. So $UV = 2 \times 28 = 56$. Consequently, $UY = YV = 28$.
3. $HZ = 25$ (assuming $W$ is typo for $H$). $HZ$ is the part of median $VZ$ from centroid to side.
* Property: Centroid divides median in 2:1 ratio. Vertex-to-Centroid : Centroid-to-Midpoint = 2:1.
* So $VH = 2 \times HZ = 2 \times 25 = 50$.
* Total median $VZ = VH + HZ = 50 + 25 = 75$.
Let's calculate the answers based on this model.
a) $ZV =$ ?
$ZV$ is the entire median from vertex $V$ to midpoint $Z$.
As calculated above, $HZ = 25$.
$VH = 2 \times HZ = 50$.
$ZV = VH + HZ = 50 + 25 = 75$.
Answer: 75
b) $ZY =$ ?
$Z$ is midpoint of $TU$. $Y$ is midpoint of $UV$.
Segment $ZY$ connects these midpoints.
By the Triangle Midsegment Theorem, $ZY$ is parallel to the third side $TV$ and is half its length.
$ZY = \frac{1}{2} TV$.
We do not know the length of side $TV$ directly from the givens ($TZ=60, XZ=28, HZ=25$).
Is there another way?
Maybe $XZ=28$ isn't a midsegment?
What if $X$ is not a midpoint? But $H$ is centroid, so lines from vertices through $H$ must hit midpoints. So $X, Y, Z$ MUST be midpoints.
Do we have enough info to find $TV$?
We know $TU = 120$.
We know $UV = 56$ (from $XZ=28$).
We know median $VZ = 75$.
We can use Apollonius theorem or the formula for median length to find side $TV$.
Formula for median $VZ$ ($m_v$) to side $v$ ($TU$):
$4 m_v^2 = 2 t^2 + 2 u^2 - v^2$ ? No, standard formula:
$m_v^2 = \frac{2 TV^2 + 2 UV^2 - TU^2}{4}$
Let $TV = x$.
$75^2 = \frac{2 x^2 + 2(56)^2 - 120^2}{4}$
$5625 = \frac{2 x^2 + 2(3136) - 14400}{4}$
$22500 = 2 x^2 + 6272 - 14400$
$22500 = 2 x^2 - 8128$
$30628 = 2 x^2$
$x^2 = 15314$
$x = \sqrt{15314} \approx 123.75$
Then $ZY = \frac{1}{2} TV \approx 61.9$.
This results in a non-integer. School problems usually have integer answers. Did I misinterpret a given?
Let's re-read the text "$WZ = 25$".
What if $W$ is the point on side $TV$? And $X$ is... something else?
But question (e) asks for $XY$.
What if the text meant $HX = 25$?
If $HX = 25$ ($X$ is midpoint of $TV$):
Then median $UX$ has part $HX=25$.
Then $UH = 50$, total median $UX = 75$.
And $XZ = 28$ is midsegment $\rightarrow UV = 56$.
$TZ = 60 \rightarrow TU = 120$.
We still need $TV$ to find $ZY$.
Finding $TV$ would require knowing more about the triangle geometry.
Let's look at the possibility that $XZ$ is NOT a midsegment.
What if $X$ is the centroid? No, $H$ is centroid.
What if the labels in the diagram are different?
Diagram 2:
Vertices $T, U, V$.
Centroid $H$.
Points on sides: Let's say $W$ on $TV$, $Y$ on $UV$, $Z$ on $TU$.
Text: $TZ=60, XZ=28, WZ=25$.
This mix of $X$ and $W$ is the key confusion.
Let's look at the questions again:
a) $ZV$
b) $ZY$
c) $ZU$
d) $TY$
e) $XY$
Questions involve $X, Y, Z$. This confirms $X, Y, Z$ are the significant points on the perimeter (midpoints).
So $W$ in the text is almost certainly a typo.
Which typo is more likely?
1. $WZ$ should be $HZ$? (Distance from centroid to midpoint Z).
2. $WZ$ should be $XZ$? (No, XZ is already given).
3. $WZ$ should be $YZ$?
4. $WZ$ should be $WX$?
If $WZ=25$ is actually $HZ=25$:
We found $ZY \approx 61.9$. Unlikely.
Let's reconsider the given $XZ = 28$.
Is it possible $X$ is not a midpoint? No, centroid definition.
Is it possible $XZ$ is not the segment between midpoints?
In the diagram, is $X$ on the median?
Sometimes problems label the intersection of the median with the side as one letter, and a point on the median as another.
But $X, Y, Z$ are consistently used for side points in such geometry problems.
Let's try another interpretation.
Maybe $W$ is the midpoint of $TV$? And $X$ is a typo for $H$?
Text: "$TZ=60, HZ=28, WZ=25$"?
If $HZ=28$:
Median $VZ$: $HZ=28 \rightarrow VH=56 \rightarrow VZ=84$.
If $W$ is midpoint of $TV$, what is $WZ$? $WZ$ is a midsegment connecting midpoints of $TV$ and $TU$.
$WZ = \frac{1}{2} UV$.
So $25 = \frac{1}{2} UV \rightarrow UV = 50$.
Then $Y$ (midpoint of $UV$) means $UY=YV=25$.
Now we have:
$TU = 120$ ($TZ=60$).
$UV = 50$.
Median $VZ = 84$.
Find side $TV$ (let's call it $w$).
$m_v^2 = \frac{2 TV^2 + 2 UV^2 - TU^2}{4}$
$84^2 = \frac{2 w^2 + 2(50)^2 - 120^2}{4}$
$7056 = \frac{2 w^2 + 5000 - 14400}{4}$
$28224 = 2 w^2 - 9400$
$37624 = 2 w^2$
$w^2 = 18812$
$w = \sqrt{18812} \approx 137.1$. Still not an integer.
Let's try: $W$ is midpoint of $TV$. Text says $WX = 28$? No, text says $XZ$.
What if $X$ is the midpoint of $TV$ and $W$ is a typo for $H$, AND $XZ=28$ is a typo for $HZ=28$?
Too many typos.
Let's look at the numbers again.
$TZ = 60$.
$XZ = 28$.
$WZ = 25$.
Is it possible that $X$ and $W$ are points on the medians?
Look at diagram 1. $P$ is centroid. $D, E, F$ are midpoints.
Look at diagram 2. $H$ is centroid.
Maybe $Z$ is a midpoint.
Maybe $X$ is a midpoint.
Maybe $W$ is a midpoint.
If $X, W, Z$ are midpoints, why does the text give $XZ$ and $WZ$? These are midsegments.
And why does it ask for $XY$? Where did $Y$ come from? $Y$ must be the third midpoint.
So the midpoints are $X, Y, Z$.
This leaves $W$ as the typo.
If $W$ is a typo for $Y$?
Given: $TZ=60, XZ=28, YZ=25$.
$Z$ midpoint of $TU \rightarrow TU=120$.
$X$ midpoint of $TV$, $Z$ midpoint of $TU \rightarrow XZ$ midsegment $\parallel UV$. $XZ = UV/2 \rightarrow UV=56$.
$Y$ midpoint of $UV$, $Z$ midpoint of $TU \rightarrow YZ$ midsegment $\parallel TV$. $YZ = TV/2 \rightarrow TV=50$.
Check consistency:
Sides are $120, 56, 50$.
Triangle Inequality: $56+50 = 106 < 120$. This triangle cannot exist!
So $YZ$ cannot be 25 if $XZ=28$ and $TZ=60$.
Okay, what if $W$ is a typo for $H$ and $XZ=28$ is correct?
We established this leads to non-integers.
What if $XZ=28$ is a typo for $HZ=28$ and $WZ=25$ is a typo for $YZ=25$?
If $HZ=28$: Median $VZ = 84$.
If $YZ=25$: Midsegment $YZ = TV/2 \rightarrow TV=50$.
Sides: $TU=120, TV=50$.
Median $VZ$ to side $TU$.
$m_v^2 =
Key Rule:
The centroid is the point where the three medians intersect. A median connects a vertex to the midpoint of the opposite side. The centroid divides each median into two parts with a specific ratio:
1. The part from the vertex to the centroid is 2/3 of the total length of the median.
2. The part from the centroid to the midpoint is 1/3 of the total length of the median.
3. Therefore, the longer part (vertex to centroid) is exactly twice the length of the shorter part (centroid to midpoint).
Let's solve for each missing measure in Problem 2 step-by-step.
Given:
* $H$ is the centroid of $\triangle TUV$.
* $TZ = 60$ ($Z$ is on side $TU$, so $VZ$ is the median from vertex $V$). Wait, looking at the diagram, $Z$ is on side $TU$. So the median is segment $VZ$. The problem states $TZ=60$. Since $VZ$ is a median, $Z$ is the midpoint of $TU$. This means $TZ = ZU$.
* $XZ = 28$ ($X$ is on side $TV$? No, let's look closer. $X$ is on side $TV$. The segment connecting $U$ to $X$ passes through $H$. So $UX$ is a median. $H$ lies on $UX$. The problem gives $XZ=28$? That seems odd. Let's re-read carefully. Ah, the text says "$XZ = 28$". Looking at the diagram, $X$ is on side $TV$ and $Z$ is on side $TU$. $XZ$ is a segment connecting two midpoints? No, usually these problems give segments along the medians. Let's look at the labels again.
* Median from $V$ goes to $Z$ on $TU$. So $V-H-Z$ is a line.
* Median from $T$ goes to $Y$ on $UV$. So $T-H-Y$ is a line.
* Median from $U$ goes to $X$ on $TV$. So $U-H-X$ is a line.
* The text says: $TZ = 60$, $XZ = 28$, and $WZ = 25$. Wait, there is no $W$ in the main triangle vertices. Let's look at the diagram for Problem 2 again. There is a point $W$ on the side $TV$? No, $W$ is on the side $TV$ in the diagram? Actually, looking at the diagram for #2:
* Vertices are $T, U, V$.
* Midpoints appear to be $X$ (on $TV$), $Y$ (on $UV$), and $Z$ (on $TU$).
* Medians are $UZ$, $TY$, and $VX$? Or $VZ$? Let's trace the lines.
* Line from $V$ goes to $Z$ on $TU$. So $VZ$ is a median.
* Line from $T$ goes to... it looks like it goes to a point on $UV$. Let's call it $Y$. So $TY$ is a median.
* Line from $U$ goes to... it looks like it goes to a point on $TV$. Let's call it $X$? But the text says $XZ=28$. If $X$ and $Z$ are midpoints, $XZ$ is a midsegment. But the question asks for lengths like $ZV$, $ZY$, etc.
* Let's re-read the text provided in the image for #2: "If $H$ is the centroid of $\triangle TUV$, $TZ = 60$, $XZ = 28$, and $WZ = 25$..."
* This text is confusing because $W$ is not a standard label. Let's look at the diagram labels very closely.
* In diagram 2:
* Vertex $T$ top left.
* Vertex $U$ bottom right? No, $U$ is top right.
* Vertex $V$ bottom.
* Point $Z$ is on side $TU$.
* Point $X$ is on side $TV$? Or is $X$ on $TU$?
* Let's look at the lines intersecting at $H$.
* One line comes from $V$, goes through $H$, to $Z$ on $TU$. So $VZ$ is a median.
* One line comes from $T$, goes through $H$, to a point on $UV$. Let's assume this point is $Y$. So $TY$ is a median.
* One line comes from $U$, goes through $H$, to a point on $TV$. Let's assume this point is $W$? Or $X$?
* The text lists $XZ=28$ and $WZ=25$. This implies $X$ and $W$ are points related to $Z$.
* Actually, let's look at the previous problem (#1) to understand the notation style. In #1, $P$ is centroid. $JK=22$ (side), $KN=13$ (part of median?), $OL=18$ (part of median?).
* Let's re-examine Problem 2's text and diagram together.
* Diagram 2 shows medians intersecting at $H$.
* The vertices are $T, U, V$.
* The points on the sides are $X, Y, Z$.
* Usually, $X$ is opposite $T$, $Y$ opposite $U$, $Z$ opposite $V$. But here:
* $Z$ is on side $TU$. So the median is from $V$ to $Z$. Thus $VZ$ is a median.
* The line from $T$ goes to side $UV$. Let's call the intersection $Y$. So $TY$ is a median.
* The line from $U$ goes to side $TV$. Let's call the intersection $X$? Or $W$?
* The text says: "$TZ = 60, XZ = 28, \text{and } WZ = 25$".
* This is extremely likely a typo in the book or my reading. Let's look at the letters again.
* Maybe the points on the sides are $W, X, Y$?
* Let's look at the diagram labels again.
* Side $TU$ has point $Z$.
* Side $UV$ has point $Y$? (Hard to see, looks like a letter near the midpoint of UV).
* Side $TV$ has point $X$? Or $W$?
* Let's assume the standard convention:
* Median from $V$ hits $TU$ at $Z$. So $Z$ is midpoint of $TU$.
* Median from $T$ hits $UV$ at $Y$. So $Y$ is midpoint of $UV$.
* Median from $U$ hits $TV$ at $X$ (or $W$). Let's assume the point on $TV$ is $W$ based on the text "$WZ=25$" being weird if $W$ is on $TV$.
* Wait, look at the text again: "$TZ = 60, XZ = 28, \text{and } WZ = 25$".
* Could $X$ and $W$ be points on the medians?
* Let's look at the questions asked:
a) $ZV =$ ?
b) $ZY =$ ?
c) $ZU =$ ?
d) $TY =$ ?
e) $XY =$ ?
* This suggests $X, Y, Z$ are the midpoints of the sides.
* If $Z$ is midpoint of $TU$, then $TZ = ZU$.
* If $Y$ is midpoint of $UV$, then $UY = YV$.
* If $X$ is midpoint of $TV$, then $TX = XV$.
* Let's re-read the given values with this hypothesis:
* "$TZ = 60$": Since $Z$ is midpoint of $TU$, $ZU = 60$ and total side $TU = 120$.
* "$XZ = 28$": $X$ and $Z$ are midpoints of sides $TV$ and $TU$. The segment connecting midpoints of two sides is parallel to the third side ($UV$) and half its length. So $XZ = \frac{1}{2} UV$. This would mean $UV = 56$. And $Y$ is midpoint of $UV$, so $UY = YV = 28$. Also $ZY$ connects midpoints of $TU$ and $UV$, so $ZY = \frac{1}{2} TV = TX = XV$. And $XY$ connects midpoints of $TV$ and $UV$, so $XY = \frac{1}{2} TU = TZ = ZU = 60$.
* "$WZ = 25$": Who is $W$? Looking at the diagram, there is a point labeled $W$ on the median from $U$? Or is $W$ the point on side $TV$? If $W$ is the point on side $TV$, then $W$ is the midpoint. So $W=X$? Why two names?
* Let's look really closely at the image text for #2. It says: "If $H$ is the centroid of $\triangle TUV$, $TZ = 60$, $XZ = 28$, and $WZ = 25$..."
* Wait, is it possible the text says $HZ = 25$? The letter 'W' and 'H' can look similar in some fonts or handwriting, but this is printed text. However, $H$ is the centroid. $Z$ is on the side. $HZ$ is the segment from centroid to side.
* Let's test the hypothesis: Given $TZ=60$, $XZ=28$, and $HZ=25$.
* If $HZ = 25$: $HZ$ is the shorter part of median $VZ$. The longer part $VH$ is $2 \times HZ = 50$. Total median $VZ = VH + HZ = 75$.
* Question (a) asks for $ZV$. $ZV$ is the whole median? Or just the segment from Z to V? Yes, $ZV$ is the median length. So $ZV = 75$.
* Question (c) asks for $ZU$. Since $VZ$ is a median, $Z$ is the midpoint of $TU$. Given $TZ = 60$, then $ZU = 60$.
* Question (b) asks for $ZY$. $Z$ is midpoint of $TU$. $Y$ is likely midpoint of $UV$. $ZY$ is a midsegment. Length $ZY = \frac{1}{2} TV$. We don't know $TV$ yet.
* What about $XZ = 28$? If $X$ is midpoint of $TV$ and $Z$ is midpoint of $TU$, then $XZ$ is a midsegment parallel to $UV$. $XZ = \frac{1}{2} UV$. So $UV = 56$. Since $Y$ is midpoint of $UV$, $UY = YV = 28$.
* Question (e) asks for $XY$. $X$ is midpoint of $TV$, $Y$ is midpoint of $UV$. $XY$ is midsegment parallel to $TU$. $XY = \frac{1}{2} TU$. Since $TU = TZ + ZU = 60 + 60 = 120$, then $XY = 60$.
* Question (d) asks for $TY$. $TY$ is the median from $T$ to side $UV$. We need the length of median $TY$. Do we have enough info? We know sides $TU=120$, $UV=56$. We don't know $TV$.
* Is there another interpretation?
* Let's look at the "W" again. In the diagram, the point on the side $TV$ is labeled $W$? Or $X$?
* Let's look at the letters in the diagram for #2.
* Top-left vertex: $T$.
* Top-right vertex: $U$.
* Bottom vertex: $V$.
* Point on $TU$: $Z$.
* Point on $UV$: $Y$? (Looks like a Y).
* Point on $TV$: $W$? (Looks like a W).
* So the midpoints are $Z, Y, W$.
* Then what is $X$? The text mentions $XZ=28$. And question (e) asks for $XY$. This implies $X$ is a point.
* Maybe the point on $TV$ is $X$? And the text has a typo saying $WZ$?
* Or maybe the point on $TV$ is $W$, and the text has a typo saying $XZ$?
* Let's look at the questions again: a) $ZV$, b) $ZY$, c) $ZU$, d) $TY$, e) $XY$.
* The questions use $X$ and $Y$. This strongly suggests the midpoints are named $X, Y, Z$.
* So, $X$ is on $TV$, $Y$ is on $UV$, $Z$ is on $TU$.
* This makes $W$ in the text "$WZ=25$" the outlier. It is highly probable that $W$ is a typo for $H$. $H$ is the centroid. $HZ$ is the distance from centroid to midpoint $Z$.
* Let's proceed with: Midpoints are $X$ (on $TV$), $Y$ (on $UV$), $Z$ (on $TU$). Centroid is $H$.
* Given:
1. $TZ = 60$. Since $Z$ is midpoint of $TU$, $ZU = 60$ and $TU = 120$.
2. $XZ = 28$. Segment connecting midpoints $X$ (of $TV$) and $Z$ (of $TU$). By Midsegment Theorem, $XZ = \frac{1}{2} UV$. So $UV = 2 \times 28 = 56$. Consequently, $UY = YV = 28$.
3. $HZ = 25$ (assuming $W$ is typo for $H$). $HZ$ is the part of median $VZ$ from centroid to side.
* Property: Centroid divides median in 2:1 ratio. Vertex-to-Centroid : Centroid-to-Midpoint = 2:1.
* So $VH = 2 \times HZ = 2 \times 25 = 50$.
* Total median $VZ = VH + HZ = 50 + 25 = 75$.
Let's calculate the answers based on this model.
a) $ZV =$ ?
$ZV$ is the entire median from vertex $V$ to midpoint $Z$.
As calculated above, $HZ = 25$.
$VH = 2 \times HZ = 50$.
$ZV = VH + HZ = 50 + 25 = 75$.
Answer: 75
b) $ZY =$ ?
$Z$ is midpoint of $TU$. $Y$ is midpoint of $UV$.
Segment $ZY$ connects these midpoints.
By the Triangle Midsegment Theorem, $ZY$ is parallel to the third side $TV$ and is half its length.
$ZY = \frac{1}{2} TV$.
We do not know the length of side $TV$ directly from the givens ($TZ=60, XZ=28, HZ=25$).
Is there another way?
Maybe $XZ=28$ isn't a midsegment?
What if $X$ is not a midpoint? But $H$ is centroid, so lines from vertices through $H$ must hit midpoints. So $X, Y, Z$ MUST be midpoints.
Do we have enough info to find $TV$?
We know $TU = 120$.
We know $UV = 56$ (from $XZ=28$).
We know median $VZ = 75$.
We can use Apollonius theorem or the formula for median length to find side $TV$.
Formula for median $VZ$ ($m_v$) to side $v$ ($TU$):
$4 m_v^2 = 2 t^2 + 2 u^2 - v^2$ ? No, standard formula:
$m_v^2 = \frac{2 TV^2 + 2 UV^2 - TU^2}{4}$
Let $TV = x$.
$75^2 = \frac{2 x^2 + 2(56)^2 - 120^2}{4}$
$5625 = \frac{2 x^2 + 2(3136) - 14400}{4}$
$22500 = 2 x^2 + 6272 - 14400$
$22500 = 2 x^2 - 8128$
$30628 = 2 x^2$
$x^2 = 15314$
$x = \sqrt{15314} \approx 123.75$
Then $ZY = \frac{1}{2} TV \approx 61.9$.
This results in a non-integer. School problems usually have integer answers. Did I misinterpret a given?
Let's re-read the text "$WZ = 25$".
What if $W$ is the point on side $TV$? And $X$ is... something else?
But question (e) asks for $XY$.
What if the text meant $HX = 25$?
If $HX = 25$ ($X$ is midpoint of $TV$):
Then median $UX$ has part $HX=25$.
Then $UH = 50$, total median $UX = 75$.
And $XZ = 28$ is midsegment $\rightarrow UV = 56$.
$TZ = 60 \rightarrow TU = 120$.
We still need $TV$ to find $ZY$.
Finding $TV$ would require knowing more about the triangle geometry.
Let's look at the possibility that $XZ$ is NOT a midsegment.
What if $X$ is the centroid? No, $H$ is centroid.
What if the labels in the diagram are different?
Diagram 2:
Vertices $T, U, V$.
Centroid $H$.
Points on sides: Let's say $W$ on $TV$, $Y$ on $UV$, $Z$ on $TU$.
Text: $TZ=60, XZ=28, WZ=25$.
This mix of $X$ and $W$ is the key confusion.
Let's look at the questions again:
a) $ZV$
b) $ZY$
c) $ZU$
d) $TY$
e) $XY$
Questions involve $X, Y, Z$. This confirms $X, Y, Z$ are the significant points on the perimeter (midpoints).
So $W$ in the text is almost certainly a typo.
Which typo is more likely?
1. $WZ$ should be $HZ$? (Distance from centroid to midpoint Z).
2. $WZ$ should be $XZ$? (No, XZ is already given).
3. $WZ$ should be $YZ$?
4. $WZ$ should be $WX$?
If $WZ=25$ is actually $HZ=25$:
We found $ZY \approx 61.9$. Unlikely.
Let's reconsider the given $XZ = 28$.
Is it possible $X$ is not a midpoint? No, centroid definition.
Is it possible $XZ$ is not the segment between midpoints?
In the diagram, is $X$ on the median?
Sometimes problems label the intersection of the median with the side as one letter, and a point on the median as another.
But $X, Y, Z$ are consistently used for side points in such geometry problems.
Let's try another interpretation.
Maybe $W$ is the midpoint of $TV$? And $X$ is a typo for $H$?
Text: "$TZ=60, HZ=28, WZ=25$"?
If $HZ=28$:
Median $VZ$: $HZ=28 \rightarrow VH=56 \rightarrow VZ=84$.
If $W$ is midpoint of $TV$, what is $WZ$? $WZ$ is a midsegment connecting midpoints of $TV$ and $TU$.
$WZ = \frac{1}{2} UV$.
So $25 = \frac{1}{2} UV \rightarrow UV = 50$.
Then $Y$ (midpoint of $UV$) means $UY=YV=25$.
Now we have:
$TU = 120$ ($TZ=60$).
$UV = 50$.
Median $VZ = 84$.
Find side $TV$ (let's call it $w$).
$m_v^2 = \frac{2 TV^2 + 2 UV^2 - TU^2}{4}$
$84^2 = \frac{2 w^2 + 2(50)^2 - 120^2}{4}$
$7056 = \frac{2 w^2 + 5000 - 14400}{4}$
$28224 = 2 w^2 - 9400$
$37624 = 2 w^2$
$w^2 = 18812$
$w = \sqrt{18812} \approx 137.1$. Still not an integer.
Let's try: $W$ is midpoint of $TV$. Text says $WX = 28$? No, text says $XZ$.
What if $X$ is the midpoint of $TV$ and $W$ is a typo for $H$, AND $XZ=28$ is a typo for $HZ=28$?
Too many typos.
Let's look at the numbers again.
$TZ = 60$.
$XZ = 28$.
$WZ = 25$.
Is it possible that $X$ and $W$ are points on the medians?
Look at diagram 1. $P$ is centroid. $D, E, F$ are midpoints.
Look at diagram 2. $H$ is centroid.
Maybe $Z$ is a midpoint.
Maybe $X$ is a midpoint.
Maybe $W$ is a midpoint.
If $X, W, Z$ are midpoints, why does the text give $XZ$ and $WZ$? These are midsegments.
And why does it ask for $XY$? Where did $Y$ come from? $Y$ must be the third midpoint.
So the midpoints are $X, Y, Z$.
This leaves $W$ as the typo.
If $W$ is a typo for $Y$?
Given: $TZ=60, XZ=28, YZ=25$.
$Z$ midpoint of $TU \rightarrow TU=120$.
$X$ midpoint of $TV$, $Z$ midpoint of $TU \rightarrow XZ$ midsegment $\parallel UV$. $XZ = UV/2 \rightarrow UV=56$.
$Y$ midpoint of $UV$, $Z$ midpoint of $TU \rightarrow YZ$ midsegment $\parallel TV$. $YZ = TV/2 \rightarrow TV=50$.
Check consistency:
Sides are $120, 56, 50$.
Triangle Inequality: $56+50 = 106 < 120$. This triangle cannot exist!
So $YZ$ cannot be 25 if $XZ=28$ and $TZ=60$.
Okay, what if $W$ is a typo for $H$ and $XZ=28$ is correct?
We established this leads to non-integers.
What if $XZ=28$ is a typo for $HZ=28$ and $WZ=25$ is a typo for $YZ=25$?
If $HZ=28$: Median $VZ = 84$.
If $YZ=25$: Midsegment $YZ = TV/2 \rightarrow TV=50$.
Sides: $TU=120, TV=50$.
Median $VZ$ to side $TU$.
$m_v^2 =
Parent Tip: Review the logic above to help your child master the concept of centroid worksheet.
