Question Video: Finding the Length of the Projection of a Triangle Side on the Straight Line Carrying Another Side Using Pythagoras’s Theorem | Nagwa Question Video: Finding the Length of the Projection of a Triangle Side on the Straight Line Carrying Another Side Using Pythagoras’s Theorem | Nagwa

# Question Video: Finding the Length of the Projection of a Triangle Side on the Straight Line Carrying Another Side Using Pythagorasβs Theorem Mathematics • Second Year of Preparatory School

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Find the length of the projection of the line segment π΄πΈ on the line π΅πΆ.

03:52

### Video Transcript

Find the length of the projection of the line segment π΄πΈ on the line passing through π΅ and πΆ.

In this question, we are asked to find the length of the projection of a line segment onto a line. To do this, letβs start by recalling what is meant by the projection of a line segment onto a line. In general, this is the line segment between the projection of the endpoints of the line segment. This is the same as saying that the projection of the line segment between π and π on the line passing through π and π is the line segment between π prime and π prime, where π prime and π prime lie on the line passing through π and π. And the line segments ππ prime and ππ prime are perpendicular with the line passing through π and π.

To project the line segment between π΄ and πΈ onto the line between π΅ and πΆ, letβs start by highlighting the two objects on the diagram. Remember, the line passing through π΅ and πΆ extends indefinitely in both directions. To project the line segment π΄πΈ onto the line passing through π΅ and πΆ, we need to project points π΄ and πΈ onto the line.

Letβs start by projecting point π΄ on the line. To project π΄ onto the line, we want to find the point on the line such that π΄π΄ prime is perpendicular to the line passing through π΅ and πΆ. We can see in the figure that the line segment between π΄ and π΅ is perpendicular to the line passing through π΅ and πΆ. So the projection of π΄ onto this line is point π΅.

We can follow a similar process to project point πΈ onto the line passing through π΅ and πΆ. We note that triangle πΆπΈπ· is an isosceles triangle. And so we can recall that the median from the base πΆπ· will be perpendicular to the base. So, πΈ prime lies at the midpoint of line segment πΆπ·. Hence, the projection of the line segment is the line segment between π΅ and πΈ prime. Remember, we want to find the length of the projection. This means we want to find π΅πΈ prime, which we can note is equal to π΅πΆ plus πΆπΈ prime. We can see in the diagram that triangle π΄π΅πΆ is a right triangle. So we can find the length of line segment π΅πΆ using the Pythagorean theorem. Its length is the square root of 58 squared minus 40 squared centimeters. We can evaluate this expression to obtain 42 centimeters.

We can see in the diagram that line segment πΆπ· has the same length as line segment π΅πΆ. And since πΈ prime is the midpoint of line segment πΆπ·, the length of line segment πΆπΈ prime is half of 42. Hence, the length of the projection of line segment π΄πΈ onto the line passing through π΅ and πΆ is 63 centimeters.

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