Question

in rhombus rstu. if ( qr = w + 13 ) and ( qt = 4w - 98 ), find ( qr ).

Explanation:

Step1: Set sides equal (rhombus property)

In a rhombus, all sides are equal, so $QR = RT$. Also, the diagonals bisect each other, so $QT = QR$ (wait, correction: actually, in a rhombus, the sides are all equal, and the diagonals bisect each other, but here we use the fact that the sides are equal, and since $QT$ is half a diagonal? No, wait, no—correction: In a rhombus, the diagonals bisect each other, but actually, the key is that all sides are equal, so $QR = ST = TU = UR$, but more importantly, the segments of the diagonals: no, wait, the problem gives $QR = w+13$ and $QT=4w-98$, and in a rhombus, the diagonals bisect each other at right angles, but actually, **the diagonals of a rhombus bisect each other, so $QT = QR$ is not right. Wait, no—wait, no, $Q$ is the intersection of the diagonals, so $QT = QU$ and $QS = QR$? No, no, in rhombus RSTU, the diagonals are $SU$ and $RT$, intersecting at $Q$. So $QT = QR$ is wrong. Wait, no! Wait, all sides of a rhombus are equal, so $QR$ is a side, $QT$ is half of diagonal $RT$. Wait, no, no—the problem says "find QR", and gives $QR = w+13$ and $QT=4w-98$. Oh! Wait, in a rhombus, the diagonals bisect each other, but also, the sides are equal, but actually, no—wait, maybe the problem has a typo? No, wait, no: in a rhombus, the diagonals bisect the angles, but no, wait, no—wait, no, actually, the key is that **the diagonals of a rhombus are perpendicular bisectors, but the segments of the diagonals are not equal to the sides unless it's a square. Wait, no, the problem must mean that $QR = QT$? No, that can't be. Wait, no—wait, no, $QR$ is a side, $QT$ is half of diagonal $RT$. Wait, no, maybe I misread: the problem says "If QR = w + 13 and QT = 4w - 98, find QR in rhombus RSTU." Oh! Wait a minute—**in a rhombus, the diagonals bisect each other, so $QT = QR$ is not true, but wait, no—wait, no, $QR$ is a side, $QT$ is a segment of the diagonal. Wait, no, maybe the problem is that in a rhombus, the diagonals are equal? No, rhombus diagonals are not equal unless it's a square. Wait, no—wait, maybe I made a mistake: the problem says $QR = w+13$ and $QT=4w-98$, and we need to set them equal? But why? Oh! Wait, no—wait, in a rhombus, the diagonals bisect each other, so $QT = RQ$? No, $QT$ is half of diagonal $RT$, $RQ$ is a side. Wait, no, maybe the problem has a typo, but actually, the only way to solve this is that the problem assumes that $QR = QT$, which would be true if the rhombus is a square, but no. Wait, no—wait, no, maybe $Q$ is a point such that $QR = QT$? No, the figure shows $Q$ is the intersection of the diagonals. Oh! Wait a second—**in a rhombus, all sides are equal, so $QR = RS = ST = TU$, and the diagonals bisect each other, so $QT = QR$ is not true, but wait, maybe the problem is that $QT = QU$, but that's $4w-98 = QU$, but we don't have $QU$. Wait, no, the problem says "find QR", so we need to find $w$ first. The only way is that the problem has a property that I'm missing: wait, no—wait, maybe the problem is that in a rhombus, the diagonals are perpendicular, so using Pythagoras? But we don't have the other diagonal. Wait, no, the problem must mean that $QR = QT$, which would be the case if the rhombus is a square, but that's not general. Wait, no—wait, maybe I misread the problem: is it $QT = 4w - 98$ and $QR = w + 13$, and in a rhombus, the diagonals bisect each other, so $QT = QR$? No, that's not a property. Wait, no—wait, maybe $Q$ is a point on the side? No, the figure shows $Q$ is the intersection of the diagonals. Oh! Wait, wait a minute—**maybe the problem is that $QR$ is a segment of the diagonal, not a side? Wait, the problem says "find QR in rhombus RSTU". Oh! Oh right! I misread: $QR$ is a segment of the diagonal, not a side. Because $Q$ is the intersection of the diagonals, so $QR$ is half of diagonal $RT$, and $QT$ is also half of diagonal $RT$? No, that would mean $QR = QT$, which would be true because diagonals bisect each other. Oh! That's it! I misidentified $QR$: $Q$ is the intersection of the diagonals, so $QR$ is half of diagonal $RT$, and $QT$ is the other half of diagonal $RT$, so they are equal. That makes sense!

Step1: Set segments equal (bisecting diagonal)

Since diagonals of a rhombus bisect each other, $QR = QT$.
$w + 13 = 4w - 98$

Step2: Solve for $w$

Rearrange to isolate $w$:
$13 + 98 = 4w - w$
$111 = 3w$
$w = \frac{111}{3} = 37$

Step3: Calculate $QR$

Substitute $w=37$ into $QR = w + 13$:
$QR = 37 + 13 = 50$

Answer:

$50$