Math

QuestionEncontre aa e bb onde as retas paralelas rr e ss são cortadas por uma transversal com a equação x2+x2+x8+8=x5+20\frac{x}{2} + \frac{x}{2} + \frac{x}{8} + 8 = \frac{x}{5} + 20.

Studdy Solution

STEP 1

Assumptions1. The relationship between the angles is given by x+x+x8+8=x5+20\frac{x}{} + \frac{x}{} + \frac{x}{8} +8 = \frac{x}{5} +20 . We are looking for the values of aa and bb in the figure3. The figure involves two parallel lines rr and ss cut by a transversal

STEP 2

First, we simplify the given equation. We combine like terms on both sides of the equation.
x2+x2+x8+8=x5+20\frac{x}{2} + \frac{x}{2} + \frac{x}{8} +8 = \frac{x}{5} +20implifying the left side, we get4x8+x8+8=x5+20\frac{4x}{8} + \frac{x}{8} +8 = \frac{x}{5} +20

STEP 3

Further simplifying the left side, we get5x8+8=x5+20\frac{5x}{8} +8 = \frac{x}{5} +20

STEP 4

To make the equation easier to solve, we multiply every term by40 (the least common multiple of8 and) to eliminate the fractions.
40x8+408=40x+402040 \cdot \frac{x}{8} +40 \cdot8 =40 \cdot \frac{x}{} +40 \cdot20

STEP 5

olving the equation, we get25x+320=8x+80025x +320 =8x +800

STEP 6

Rearranging the equation to isolate xx, we get25x8x=80032025x -8x =800 -320

STEP 7

olving for xx, we get17x=48017x =480

STEP 8

Dividing both sides by17 to solve for xx, we getx=48017x = \frac{480}{17}

STEP 9

olving the fraction, we getx=28.235294117647058x =28.235294117647058

STEP 10

Now that we have xx, we can find the values of aa and bb using the properties of parallel lines cut by a transversal.For parallel lines cut by a transversal, corresponding angles are equal and alternate interior angles are equal.
So, we havea=x2a = \frac{x}{2}b=x2b = \frac{x}{2}

STEP 11

Substituting the value of xx into the equations for aa and bb, we geta=28.235294117647058a = \frac{28.235294117647058}{}b=28.235294117647058b = \frac{28.235294117647058}{}

STEP 12

olving for aa and bb, we geta=14.117647058823529a =14.117647058823529b=14.117647058823529b =14.117647058823529Therefore, the values of aa and bb are approximately14.12.

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