**Parallelogram BMVP** is given, which implies opposite sides are equal.

**Given Equalities:**

BM=PV

BP=MV

**Expressions for sides:**

BM=3x+12

MV=2y−8

PV=5y+34

BP=x−3

Thank you for the clarification. Let's solve the problem using the correct expressions and relationships for the parallelogram BMVP.

### Given Information:

1. **Parallelogram Properties**: In parallelogram BMVP:
   - \( BM = PV \)
   - \( BP = MV \)

2. **Expressions for sides**:
   - \( BM = 3x + 12 \)
   - \( MV = 2y - 8 \)
   - \( PV = 5y + 34 \)
   - \( BP = x - 3 \)

### Step 1: Set up equations using the properties of the parallelogram

Given the properties \( BM = PV \) and \( BP = MV \):

#### Equation 1: Equating \( BM \) and \( PV \)
\[
3x + 12 = 5y + 34
\]

#### Equation 2: Equating \( BP \) and \( MV \)
\[
x - 3 = 2y - 8
\]

### Step 2: Solve for \( x \) and \( y \)

Start by simplifying both equations.

**Simplify Equation 1:**
\[
3x - 5y = 34 - 12
\]
\[
3x - 5y = 22 \quad (1)
\]

**Simplify Equation 2:**
\[
x - 2y = -8 + 3
\]
\[
x - 2y = -5 \quad (2)
\]

### Step 3: Solve the simultaneous equations

We can solve these two equations simultaneously. Let's solve for \( x \) in terms of \( y \) using Equation 2 and then substitute into Equation 1.

From Equation (2):
\[
x = 2y - 5
\]

Substitute \( x = 2y - 5 \) into Equation (1):
\[
3(2y - 5) - 5y = 22
\]
\[
6y - 15 - 5y = 22
\]
\[
y - 15 = 22
\]
\[
y = 37
\]

Now substitute \( y = 37 \) back into the expression for \( x \):
\[
x = 2(37) - 5
\]
\[
x = 74 - 5
\]
\[
x = 69
\]

### Step 4: Calculate the side lengths

Now that we have the values for \( x \) and \( y \):

- **\( BM \) (or \( PV \)):**
\[
BM = 3x + 12 = 3(69) + 12 = 207 + 12 = 219
\]

- **\( PV \) (or \( BM \)):**
\[
PV = 5y + 34 = 5(37) + 34 = 185 + 34 = 219
\]

- **\( BP \) (or \( MV \)):**
\[
BP = x - 3 = 69 - 3 = 66
\]

- **\( MV \) (or \( BP \)):**
\[
MV = 2y - 8 = 2(37) - 8 = 74 - 8 = 66
\]

### Step 5: Calculate the perimeter of the parallelogram

The perimeter \( P \) of a parallelogram is given by the formula:
\[
P = 2(BM + BP)
\]

Substitute the calculated values:
\[
P = 2(219 + 66) = 2(285) = 570
\]

### Final Answer:
The perimeter of parallelogram BMVP is **570 units**.

Would you like any further explanation, or do you have any questions?

Here are some follow-up questions you might find interesting:

1. How can different expressions for sides of a parallelogram affect the solution?
2. What are the properties of a parallelogram that are crucial for solving such problems?
3. How would the solution change if BMVP were a rhombus or rectangle instead of a general parallelogram?
4. What is the significance of understanding linear equations when dealing with geometric shapes?
5. Can you think of any other methods to solve the given problem?

**Tip:** When solving problems involving algebraic expressions, carefully substitute values back into all equations to ensure consistency and accuracy.

Learn how to solve a parallelogram problem involving sides BM, MV, PV, and BP with given equalities and expressions. This detailed solution covers key geometric concepts and linear equations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation