# Question Video: Geometric Applications of Vectors in a Trapezium Mathematics

π΄π΅πΆπ· is a trapezium. If ππ + ππ = ππ±π², then π = οΌΏ, where π β β. [A] β2 [B] β1 [C] 1 [D] 2

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### Video Transcript

π΄π΅πΆπ· is a trapezium. If vector ππ plus vector ππ is equal to π multiplied by vector π±π², then π is equal to blank, where π is a real number. Is the answer (A) negative two, (B) negative one, (C) one, or (D) two?

We know that two vectors are equal if they have the same direction and magnitude. As π± is the midpoint of line segment ππ, then the vector ππ± is equal to the vector π±π. And both of these are equal to a half multiplied by the vector ππ. In the same way, as π² is the midpoint of the line segment ππ, then vector ππ² is equal to vector π²π. And both of these are equal to a half of the vector ππ.

We know that the general vectors ππ and ππ are the additive inverse of each other. This means that the vector ππ plus the vector ππ is equal to the zero vector. This can be rewritten such that the vector ππ is equal to the negative of vector ππ.

Letβs now consider the equation we are given in this question in order to calculate π. We can see from the diagram that the vector ππ is equal to the vector ππ± plus the vector π±π² plus the vector π²π. In the same way, the vector ππ is equal to the vector ππ± plus the vector π±π² plus the vector π²π. We can then replace the vector ππ± with the vector π±π and the vector π²π with the vector ππ². Collecting like terms, we have two π±π² plus π±π plus ππ± plus π²π plus ππ². Vectors π±π and ππ± as well as vectors π²π and ππ² are additive inverses. This means they are equal to the zero vector and will therefore cancel.

Our expression simplifies to two multiplied by the vector π±π². This is in the form we were looking for. And we can see that π is equal to two. The correct answer is therefore option (D). If π΄π΅πΆπ· is a trapezium and vector ππ plus vector ππ is equal to π multiplied by vector π±π², then π is equal to two.