Questions: If QR=QT=45, RS=u+42, and ST=8u, what is the value of u ? u=

Transcript text: If $QR=QT=45, RS=u+42$, and $ST=8u$, what is the value of $u ?$ \[ u= \]

Solution

Solution Steps

Step 1: Identify the Given Information

$ QR = QT = 45 $
$ RS = u + 42 $
$ ST = 8u $

Step 2: Use the Pythagorean Theorem

Since $ \triangle QRS $ is a right triangle at $ S $, we can use the Pythagorean theorem: \[ QR^2 = QS^2 + RS^2 \]

Step 3: Express $ QS $ in Terms of $ u $

Since $ QT = 45 $ and $ ST = 8u $, we can express $ QS $ as: \[ QS = QT - ST = 45 - 8u \]

Step 4: Substitute and Solve for $ u $

Substitute $ QR = 45 $, $ QS = 45 - 8u $, and $ RS = u + 42 $ into the Pythagorean theorem: \[ 45^2 = (45 - 8u)^2 + (u + 42)^2 \]

Step 5: Simplify the Equation

\[ 2025 = (45 - 8u)^2 + (u + 42)^2 \] \[ 2025 = (2025 - 720u + 64u^2) + (u^2 + 84u + 1764) \] \[ 2025 = 65u^2 - 636u + 3789 \]

Step 6: Rearrange and Solve the Quadratic Equation

\[ 0 = 65u^2 - 636u + 1764 \] Use the quadratic formula $ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $: \[ a = 65, b = -636, c = 1764 \] \[ u = \frac{636 \pm \sqrt{636^2 - 4 \cdot 65 \cdot 1764}}{2 \cdot 65} \] \[ u = \frac{636 \pm \sqrt{404496 - 458640}}{130} \] \[ u = \frac{636 \pm \sqrt{-54144}}{130} \]

Since the discriminant is negative, there is no real solution for $ u $. Therefore, we need to recheck the problem setup or calculations.

Final Answer

There is no real value for $ u $ that satisfies the given conditions.

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