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

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

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Given that π΄π΅ = 29, πΆπ΅ = 20, and πΆπ· = 35, calculate the length of the projection of line segment πΆπ· on line π΄π·.

03:43

### Video Transcript

Given that π΄π΅ equals 29, πΆπ΅ equals 20, and πΆπ· equals 35, calculate the length of the projection of line segment πΆπ· on line π΄π·.

Before we can calculate the length of this projection, weβll need to see which line segment this projection is. Line π΄π· is our target line. Thatβs the line upon which the projection will fall. And πΆπ· is the line segment that weβll use to create our projection. But in order to create a projection, weβll need perpendicular lines to our target line. Because we know that line π΅πΆ and line π΄π· are parallel, the angle π΅πΆπ΄ is an alternate interior angle to the angle πΆπ΄π·, which means the line segment π΄πΆ is perpendicular to the line segment π΄π·.

Once we imagine the light source as the set of all the perpendicular lines to our target line, we can find the endpoints of our projection. Starting at πΆ, the projection of point πΆ onto π΄π· would be the point π΄ and the projection of point π· onto π΄π· would be itself. And that means the projection of πΆπ· is going to be equal to π΄π·. This is the value we want to calculate the length of. To do that, letβs clear out our light source lines and think about what we know about right triangles.

First of all, we know that π΄π΅ has a measure of 29 and πΆπ΅ has a measure of 20 and πΆπ· has a measure of 35. What we have in our figure is two separate right triangles that make up this quadrilateral. And so we remember that we can find side lengths in right triangles using the Pythagorean theorem, where π and π represent the two smaller sides and π represents the hypotenuse. π squared plus π squared equals π squared. To use the Pythagorean theorem, you need at least two of the lengths. Since we donβt know the length of π΄πΆ, we canβt find the length of π΄π·. However, we can use the information about the smaller triangle π΄π΅πΆ to solve for the side length π΄πΆ first.

When we plug in what we know, we get 20 squared plus π΄πΆ squared equals 29 squared. 400 plus π΄πΆ squared equals 841. So weβll subtract 400 from both sides of the equation, and weβll get π΄πΆ squared equals 441. After that, weβll take the square root of both sides. Weβre only interested in the positive square root since weβre dealing with distance, and so we see that π΄πΆ equals 21. Now that we know that π΄πΆ is 21, we know two distances in our right triangle and weβll be able to find the side length of our third distance π΄π·. But weβll need to set up the Pythagorean theorem for a second time.

This time, weβll have 21 squared plus π΄π· squared equals 35 squared. 441 plus 80 squared equals 1225. And so we subtract 441 from both sides, and weβll get π΄π· squared equals 784. So we take the square root of both sides. Again, weβre only interested in the positive square root of 784, which is 28. The line segment π΄π· will be equal to 28. The line segment π΄π· is the projection of the line segment πΆπ· onto line π΄π·, and it has a measure of 28.

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