Question Video: Using Vectors to Find the Area of a Right Trapezoid | Nagwa Question Video: Using Vectors to Find the Area of a Right Trapezoid | Nagwa

# Question Video: Using Vectors to Find the Area of a Right Trapezoid Mathematics • First Year of Secondary School

## Join Nagwa Classes

Trapezoid π΄π΅πΆπ· has vertices π΄(4, 14), π΅(4, β4), πΆ(β12, β4), and π·(β12, 9). Given that ππ β₯ ππ and ππ β₯ ππ, find the area of that trapezoid.

03:30

### Video Transcript

Trapezoid π΄π΅πΆπ· has vertices π΄: four, 14; π΅: four, negative four; πΆ: negative 12, negative four; and π·: negative 12, nine. Given that vector ππ is parallel to vector ππ and vector ππ is perpendicular to vector ππ, find the area of that trapezoid.

Letβs consider what we are told about the trapezoid. We know that vector ππ is parallel to vector ππ. We also know that vector ππ is perpendicular to vector ππ. This means that they meet at right angles. It therefore follows that vector ππ also meets vector ππ at right angles. We know that the area of a trapezoid can be found using the formula π plus π over two multiplied by β, where π and π are the parallel sides and β is the perpendicular height. We need to find the length of the sides π΄π΅ and π·πΆ and the perpendicular height πΆπ΅.

In order to calculate the length of the sides, we need to work out the magnitude of the vectors. We will begin by calculating the magnitude of ππ. This is equal to the square root of four minus four squared plus negative four minus 14 squared. Four minus four is equal to zero. And negative four minus 14 is negative 18. Squaring this gives us 324, and then square rooting the answer gives us 18. The magnitude of vector ππ is 18. We can repeat this process to calculate the magnitude of vector ππ. This is equal to the square root of negative 12 minus negative 12 squared plus negative four minus nine squared. This gives us an answer of 13.

As the magnitude of ππ is greater than the magnitude of ππ, we can see that our sketch has not been drawn to scale. It would therefore make more sense to relabel it as shown. Vector ππ is still parallel to vector ππ, and vector ππ is perpendicular to vector ππ. We can now add the lengths onto our diagram. We now need to calculate the magnitude of vector ππ. Using the same method, we see that this is equal to 16. We now have the lengths of the parallel sides as well as the length of the perpendicular height of the trapezoid. The area is therefore equal to 13 plus 18 divided by two multiplied by 16. 13 plus 18 is equal to 31. Multiplying 31 over two by 16 gives us 248. The area of the trapezoid is therefore equal to 248 square units.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions