Question Video: Applying the Pythagorean Theorem to Solve More Complex Problems | Nagwa Question Video: Applying the Pythagorean Theorem to Solve More Complex Problems | Nagwa

# Question Video: Applying the Pythagorean Theorem to Solve More Complex Problems Mathematics • First Year of Preparatory School

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Find the perimeter of π΄π΅πΆπ·.

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### Video Transcript

Find the perimeter of π΄π΅πΆπ·.

Weβve been asked to find the perimeter of this quadrilateral, which is the sum of all its side lengths. Thatβs π΄π΅ plus π΅πΆ plus πΆπ· plus π·π΄. Weβve been given the lengths of three of these sides, but one side length, π·π΄, is unknown. Weβll need to use the information weβve been given to calculate this length.

Looking more closely at the figure, we can see that the quadrilateral π΄π΅πΆπ· has been divided into two right triangles: triangle π΄π΅πΆ and triangle π΄πΆπ·. We can calculate the length of any side in a right triangle if we know the lengths of the other two sides by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. If we label the two shorter sides as π and π and the hypotenuse as π, then this can be expressed as π squared plus π squared equals π squared. But in the pink triangle, in which side π·π΄ is the hypotenuse, weβre only given the length of one of the other sides. Side length π΄πΆ is also unknown.

However, this side is common with triangle π΄π΅πΆ in which we do know both of the other two side lengths. So, our approach will be to first calculate the length of π΄πΆ using the Pythagorean theorem in triangle π΄π΅πΆ and then use this to apply the Pythagorean theorem a second time in triangle π΄πΆπ·. π΄πΆ is the hypotenuse of triangle π΄π΅πΆ, so applying the Pythagorean theorem gives the equation π΄πΆ squared equals 20 squared plus 48 squared. Evaluating each of the squares and finding their sum gives π΄πΆ squared equals 2704.

Now, we could continue and find the length of π΄πΆ by square rooting, but actually we donβt need to. When we apply the Pythagorean theorem in triangle π΄πΆπ·, in which π·π΄ is the hypotenuse, we obtain the equation π·π΄ squared equals πΆπ· squared plus π΄πΆ squared. We know πΆπ· is 39 centimeters, and weβve already found that the squared value of π΄πΆ is 2704. So rather than square rooting to find π΄πΆ and then squaring again in our second application of the Pythagorean theorem, we can just substitute the value of π΄πΆ squared directly. That gives π·π΄ squared equals 39 squared plus 2704. Evaluating 39 squared gives 1521, and then adding 2704 gives 4225. Taking the square root of both sides of this equation gives π·π΄ equals 65. Weβre only interested in the positive solution here as π·π΄ is a length.

So, having calculated the length of the final side in quadrilateral π΄π΅πΆπ·, weβre now ready to find its perimeter. Substituting the value of 65 into our sum for the perimeter gives 20 plus 48 plus 39 plus 65, which is 172. By applying the Pythagorean theorem twice, weβve found the length of the final side of this quadrilateral and hence its perimeter, which is 172 centimeters.

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