Math / Geometry

Question $\text{In parallelogram } \mathrm{LMNO}, \text{ given } \mathrm{MP}=21 \mathrm{~m}, \mathrm{LP}=(y+3) \mathrm{m}, \mathrm{NP}=(3y-1) \mathrm{m}, \text{ and } P=(2x-1) \mathrm{m}.$ \text{What are the values of } x \text{ and } y \text{ such that:} \begin{align} x &= 10 \mathrm{~m}, y = 1 \mathrm{~m} \\ x &= 10 \mathrm{~m}, y = 2 \mathrm{~m} \\ x &= 11 \mathrm{~m}, y = 1 \mathrm{~m} \\ x &= 11 \mathrm{~m}, y = 2 \mathrm{~m} \end{align}

Studdy Solution

STEP 1

1. The point $P$ is inside the parallelogram $\mathrm{LMNO}$ .
2. The segments $\mathrm{LP}$ , $\mathrm{NP}$ , and $\mathrm{OP}$ are given in terms of $y$ and $x$ .
3. The length of $\mathrm{MP}$ is $21$ meters.

STEP 2

1. Set up equations based on the given segment lengths.
2. Solve for $y$ .
3. Solve for $x$ .
4. Verify solutions with the given options.

STEP 3

Set up equations based on the given segment lengths:
Since $\mathrm{MP} = 21$ m, we have:
$\mathrm{MP} = \mathrm{LP} + \mathrm{NP} + \mathrm{OP}$
Substitute the given expressions:
$21 = (y + 3) + (3y - 1) + (2x - 1)$
Simplify the equation:
$21 = y + 3 + 3y - 1 + 2x - 1$ $21 = 4y + 2x + 1$
Rearrange the equation:
$20 = 4y + 2x$

STEP 4

Solve for $y$ :
Rearrange the equation to express $y$ in terms of $x$ :
$4y = 20 - 2x$ $y = \frac{20 - 2x}{4}$ $y = 5 - \frac{x}{2}$

STEP 5

Solve for $x$ :
Substitute the values of $x$ from the options into the equation for $y$ to find corresponding $y$ .
Option 1: $x = 10$
$y = 5 - \frac{10}{2}$ $y = 5 - 5$ $y = 0$
Option 2: $x = 11$
$y = 5 - \frac{11}{2}$ $y = 5 - 5.5$ $y = -0.5$

STEP 6

Verify solutions with the given options:
The calculated values of $y$ do not match any of the given options, indicating a possible error in the problem setup or assumptions.

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Studdy Solution

STEP 1

1. The point $P$ is inside the parallelogram $\mathrm{LMNO}$ .
2. The segments $\mathrm{LP}$ , $\mathrm{NP}$ , and $\mathrm{OP}$ are given in terms of $y$ and $x$ .
3. The length of $\mathrm{MP}$ is $21$ meters.

STEP 2

1. Set up equations based on the given segment lengths.
2. Solve for $y$ .
3. Solve for $x$ .
4. Verify solutions with the given options.

STEP 3

STEP 4

Solve for $y$ :
Rearrange the equation to express $y$ in terms of $x$ :
$4y = 20 - 2x$ $y = \frac{20 - 2x}{4}$ $y = 5 - \frac{x}{2}$

STEP 5

Solve for $x$ :
Substitute the values of $x$ from the options into the equation for $y$ to find corresponding $y$ .
Option 1: $x = 10$
$y = 5 - \frac{10}{2}$ $y = 5 - 5$ $y = 0$
Option 2: $x = 11$
$y = 5 - \frac{11}{2}$ $y = 5 - 5.5$ $y = -0.5$

STEP 6

Verify solutions with the given options:
The calculated values of $y$ do not match any of the given options, indicating a possible error in the problem setup or assumptions.

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Studdy Solution

STEP 1

1. The point P P P is inside the parallelogram LMNO \mathrm{LMNO} LMNO.2. The segments LP \mathrm{LP} LP, NP \mathrm{NP} NP, and OP \mathrm{OP} OP are given in terms of y y y and x x x.3. The length of MP \mathrm{MP} MP is 21 21 21 meters.

STEP 2

1. Set up equations based on the given segment lengths.2. Solve for y y y.3. Solve for x x x.4. Verify solutions with the given options.

STEP 3

STEP 4

Solve for y y y:Rearrange the equation to express y y y in terms of x x x:4y=20−2x 4y = 20 - 2x 4y=20−2x y=20−2x4 y = \frac{20 - 2x}{4} y=420−2x​ y=5−x2 y = 5 - \frac{x}{2} y=5−2x​

STEP 5

STEP 6

Verify solutions with the given options:The calculated values of y y y do not match any of the given options, indicating a possible error in the problem setup or assumptions.

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1. The point $P$ is inside the parallelogram $\mathrm{LMNO}$ .
2. The segments $\mathrm{LP}$ , $\mathrm{NP}$ , and $\mathrm{OP}$ are given in terms of $y$ and $x$ .
3. The length of $\mathrm{MP}$ is $21$ meters.

1. Set up equations based on the given segment lengths.
2. Solve for $y$ .
3. Solve for $x$ .
4. Verify solutions with the given options.

Solve for $y$ :
Rearrange the equation to express $y$ in terms of $x$ :
$4y = 20 - 2x$ $y = \frac{20 - 2x}{4}$ $y = 5 - \frac{x}{2}$

Verify solutions with the given options:
The calculated values of $y$ do not match any of the given options, indicating a possible error in the problem setup or assumptions.