Question Video: Finding the Width of a Parallelogram given the Perimeter and a Relation between the Parallelogram’s Dimensions | Nagwa Question Video: Finding the Width of a Parallelogram given the Perimeter and a Relation between the Parallelogram’s Dimensions | Nagwa

Question Video: Finding the Width of a Parallelogram given the Perimeter and a Relation between the Parallelogramβs Dimensions

π΄π΅πΆπ· is a parallelogram. Given that the perimeter of π΄π΅πΆπ· is 54, find the length of π΄π΅.

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Video Transcript

π΄π΅πΆπ· is a parallelogram. Given that the perimeter of π΄π΅πΆπ· is 54, find the length of π΄π΅.

Letβs look at the diagram carefully. The lengths of the four sides of the parallelogram have been given in terms of three variables: π₯, π¦, and π§.

In order to work out the length of π΄π΅, weβre going to need to know the values of these variables or at least the value of π¦. Weβre told that the perimeter of the parallelogram π΄π΅πΆπ· is 54, so letβs start with that. The perimeter of a parallelogram is the sum of all four of its sides. So this means if we add together the expressions for π΄π΅, π΅πΆ, πΆπ·, and π·π΄, in terms of π₯, π¦, and π§, we must get 54.

This gives the equation on screen. Now this equation can be simplified by grouping like terms together. Thereβs only one π¦-term, so thatβs still seven π¦. The three π₯ and the five π₯ make eight π₯, and the seven π§ and six π§ make 13π§. The constant on the left-hand side also simplifies to negative 53. We can then add 53 to both sides of this equation, giving us seven π¦ plus eight π₯ plus 13π§ equals 107.

Now this is one equation with three unknown letters: π₯, π¦, and π§. So there isnβt anything else we can do with this equation at this stage. Letβs look back to the diagram to see what else we can do. A key fact about parallelograms is that opposite sides are of equal length. This means that π΄π΅ is equal to πΆπ·, and also π΄π· is equal to π΅πΆ.

Letβs substitute the expressions for each of these sides in terms of the letters π₯, π¦, and π§. For the first pair of sides, we have that seven π¦ minus 38 is equal to six π§ minus one. Adding 38 to both sides simplifies this to seven π¦ equals six π§ plus 37. Now this is one equation with two letters so thereβs nothing further we can do with this. Letβs look at the equations for π΄π· and π΅πΆ. Substituting the expressions for each side gives five π₯ minus four is equal to three π₯ plus seven π§ minus 10.

Subtracting three π₯ from both sides and also adding four simplifies this equation to two π₯ is equal to seven π§ minus six. Now we canβt solve either these equations, but what theyβve done is give us both π¦ and π₯ in terms of π§. Each of these expressions for π₯ and π¦ can be substituted into our equation for the perimeter so that instead of it involving π₯, π¦, and π§, it now just involves π§.

The expression six π§ plus 37 can be substituted for the seven π¦ in the equation. If we think of eight π₯ as four multiplied by two π₯, then our expression for two π₯ of seven π§ minus six can also be substituted into the equation for the perimeter. Making both of these substitutions gives six π§ plus 37 plus four lots of seven π§ minus six plus 13π§ is equal to 107.

Expanding brackets and grouping like terms gives 47π§ plus 13 equals 107. Subtracting 13 from both sides of this equation gives 47π§ equals 94. Finally, dividing both sides of the equation by 47 tells us that the value of π§ is two. Now we could go on to use this value of π§ to calculate the values of π₯ and π¦, but letβs look back at the question.

The question asked us to find the length of π΄π΅. Weβve already stated that the length of π΄π΅ is the same as the length of πΆπ·. We have an expression for the length of πΆπ· in terms of π§, and therefore we can use this to work out the length of π΄π΅. So substituting the known value of π§, which is two, into this expression tells us that the length of π΄π΅ is six multiplied by two minus one, which is 11, and therefore we have our answer to the problem: the length of π΄π΅ is 11.

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