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 Pythagoras’s Theorem Mathematics • Second Year of Preparatory School

Join Nagwa Classes

Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

Find the length of the projection of the line segment 𝐴𝐸 on the line 𝐵𝐶.

03:52

Video Transcript

Find the length of the projection of the line segment 𝐴𝐸 on the line passing through 𝐵 and 𝐶.

In this question, we are asked to find the length of the projection of a line segment onto a line. To do this, let’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 𝑃 and 𝑄 on the line passing through 𝑅 and 𝑆 is the line segment between 𝑃 prime and 𝑄 prime, where 𝑃 prime and 𝑄 prime lie on the line passing through 𝑅 and 𝑆. And the line segments 𝑃𝑃 prime and 𝑄𝑄 prime are perpendicular with the line passing through 𝑅 and 𝑆.

To project the line segment between 𝐴 and 𝐸 onto the line between 𝐵 and 𝐶, let’s start by highlighting the two objects on the diagram. Remember, the line passing through 𝐵 and 𝐶 extends indefinitely in both directions. To project the line segment 𝐴𝐸 onto the line passing through 𝐵 and 𝐶, we need to project points 𝐴 and 𝐸 onto the line.

Let’s start by projecting point 𝐴 on the line. To project 𝐴 onto the line, we want to find the point on the line such that 𝐴𝐴 prime is perpendicular to the line passing through 𝐵 and 𝐶. We can see in the figure that the line segment between 𝐴 and 𝐵 is perpendicular to the line passing through 𝐵 and 𝐶. So the projection of 𝐴 onto this line is point 𝐵.

We can follow a similar process to project point 𝐸 onto the line passing through 𝐵 and 𝐶. We note that triangle 𝐶𝐸𝐷 is an isosceles triangle. And so we can recall that the median from the base 𝐶𝐷 will be perpendicular to the base. So, 𝐸 prime lies at the midpoint of line segment 𝐶𝐷. Hence, the projection of the line segment is the line segment between 𝐵 and 𝐸 prime. Remember, we want to find the length of the projection. This means we want to find 𝐵𝐸 prime, which we can note is equal to 𝐵𝐶 plus 𝐶𝐸 prime. We can see in the diagram that triangle 𝐴𝐵𝐶 is a right triangle. So we can find the length of line segment 𝐵𝐶 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 𝐶𝐷 has the same length as line segment 𝐵𝐶. And since 𝐸 prime is the midpoint of line segment 𝐶𝐷, the length of line segment 𝐶𝐸 prime is half of 42. Hence, the length of the projection of line segment 𝐴𝐸 onto the line passing through 𝐵 and 𝐶 is 63 centimeters.

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy