Math / Geometry

QuestionGiven the $\square$ JKLM, Find the value for $x=$ $\qquad$ $Y=$ $\qquad$ $z=$ $\qquad$

Studdy Solution

STEP 1

What is this asking? We need to find the values of x, y, and z in a parallelogram, knowing the lengths of its sides and parts of its diagonals. Watch out! Remember that opposite sides of a parallelogram are equal, and its diagonals bisect each other.
Don't mix these up!

STEP 2

1. Solve for z
2. Solve for x
3. Solve for y

STEP 3

Alright, let's start with z!
We know that opposite sides of a parallelogram are equal.
Looking at sides JK and ML, we can write: $JK = ML$ .

STEP 4

Now, substitute the given expressions: $16z - 4 = 8z + 4$ .

STEP 5

Let's isolate z.
Add 4 to both sides: $16z = 8z + 8$ .

STEP 6

Subtract $8z$ from both sides: $8z = 8$ .

STEP 7

Divide both sides by 8: $z = 1$ .
Boom! We found z!

STEP 8

Next up, x!
We know that opposite sides KL and JM are equal, so $KL = JM$ .

STEP 9

We also know that $ML = JK$ , and since we found $z = 1$ , we can find the length of JK: $JK = 16 \cdot 1 - 4 = 12$ .
Since $ML = JK$ , then $ML = 12$ .

STEP 10

We're given that $ML = 2x - 2$ , so we can write $12 = 2x - 2$ .

STEP 11

Let's solve for x.
Add 2 to both sides: $14 = 2x$ .

STEP 12

Divide both sides by 2: $x = 7$ .
Yes! x is conquered!

STEP 13

Finally, y!
Remember that the diagonals of a parallelogram bisect each other.
This means $KN = NL$ .

STEP 14

We know $x = 7$ , so $KN = 7 + 6 = 13$ .

STEP 15

Since $KN = NL$ , we have $13 = 3y + 2$ .

STEP 16

Let's wrap this up!
Subtract 2 from both sides: $11 = 3y$ .

STEP 17

Divide both sides by 3: $y = \frac{11}{3}$ .
We did it!

STEP 18

$x = 7$ , $y = \frac{11}{3}$ , and $z = 1$ .

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QuestionGiven the □\square□ JKLM, Find the value for x=x=x= \qquad Y=Y=Y= \qquad z=z=z= \qquad

Studdy Solution

STEP 1

What is this asking? We need to find the values of *x*, *y*, and *z* in a parallelogram, knowing the lengths of its sides and parts of its diagonals. Watch out! Remember that opposite sides of a parallelogram are equal, and its diagonals bisect each other.Don't mix these up!

STEP 2

1. Solve for *z*2. Solve for *x*3. Solve for *y*

STEP 3

Alright, let's **start** with *z*!We know that opposite sides of a parallelogram are equal.Looking at sides JK and ML, we can write: JK=MLJK = MLJK=ML.

STEP 4

Now, **substitute** the given expressions: 16z−4=8z+416z - 4 = 8z + 416z−4=8z+4.

STEP 5

Let's **isolate** *z*.Add 4 to both sides: 16z=8z+816z = 8z + 816z=8z+8.

STEP 6

Subtract 8z8z8z from both sides: 8z=88z = 88z=8.

STEP 7

**Divide** both sides by 8: z=1z = 1z=1.Boom! We found *z*!

STEP 8

Next up, *x*!We know that opposite sides KL and JM are equal, so KL=JMKL = JMKL=JM.

STEP 9

We also know that ML=JKML = JKML=JK, and since we found z=1z = 1z=1, we can find the length of JK: JK=16⋅1−4=12JK = 16 \cdot 1 - 4 = 12JK=16⋅1−4=12.Since ML=JKML = JKML=JK, then ML=12ML = 12ML=12.

STEP 10

We're given that ML=2x−2ML = 2x - 2ML=2x−2, so we can write 12=2x−212 = 2x - 212=2x−2.

STEP 11

Let's **solve for** *x*.Add 2 to both sides: 14=2x14 = 2x14=2x.

STEP 12

**Divide** both sides by 2: x=7x = 7x=7.Yes! *x* is conquered!

STEP 13

Finally, *y*!Remember that the diagonals of a parallelogram bisect each other.This means KN=NLKN = NLKN=NL.

STEP 14

We know x=7x = 7x=7, so KN=7+6=13KN = 7 + 6 = 13KN=7+6=13.

STEP 15

Since KN=NLKN = NLKN=NL, we have 13=3y+213 = 3y + 213=3y+2.

STEP 16

Let's **wrap this up**!Subtract 2 from both sides: 11=3y11 = 3y11=3y.

STEP 17

**Divide** both sides by 3: y=113y = \frac{11}{3}y=311​.We did it!

STEP 18

x=7x = 7x=7, y=113y = \frac{11}{3}y=311​, and z=1z = 1z=1.

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QuestionGiven the $\square$ JKLM, Find the value for $x=$ $\qquad$ $Y=$ $\qquad$ $z=$ $\qquad$

What is this asking? We need to find the values of x, y, and z in a parallelogram, knowing the lengths of its sides and parts of its diagonals. Watch out! Remember that opposite sides of a parallelogram are equal, and its diagonals bisect each other.
Don't mix these up!

1. Solve for z
2. Solve for x
3. Solve for y

Alright, let's start with z!
We know that opposite sides of a parallelogram are equal.
Looking at sides JK and ML, we can write: $JK = ML$ .

Now, substitute the given expressions: $16z - 4 = 8z + 4$ .

Let's isolate z.
Add 4 to both sides: $16z = 8z + 8$ .

Subtract $8z$ from both sides: $8z = 8$ .

Divide both sides by 8: $z = 1$ .
Boom! We found z!

Next up, x!
We know that opposite sides KL and JM are equal, so $KL = JM$ .

We also know that $ML = JK$ , and since we found $z = 1$ , we can find the length of JK: $JK = 16 \cdot 1 - 4 = 12$ .
Since $ML = JK$ , then $ML = 12$ .

We're given that $ML = 2x - 2$ , so we can write $12 = 2x - 2$ .

Let's solve for x.
Add 2 to both sides: $14 = 2x$ .

Divide both sides by 2: $x = 7$ .
Yes! x is conquered!

Finally, y!
Remember that the diagonals of a parallelogram bisect each other.
This means $KN = NL$ .

We know $x = 7$ , so $KN = 7 + 6 = 13$ .

Since $KN = NL$ , we have $13 = 3y + 2$ .

Let's wrap this up!
Subtract 2 from both sides: $11 = 3y$ .

Divide both sides by 3: $y = \frac{11}{3}$ .
We did it!

$x = 7$ , $y = \frac{11}{3}$ , and $z = 1$ .