Looks Simple, But Most People Fail! Find the Length X! |

[00:00:00]Hello everyone and welcome back to my channel.

[00:00:03]In today's video, we are going to solve another interesting geometric problem.

[00:00:08]In the given question, PQRS is a rectangle.

[00:00:14]Line segment TPS is a straight line extending from diagonal PS such that TP is equal to eight units and PS is equal to 10 units.

[00:00:26]Angle QPS is equal to angle TRP and TR is equal to X.

[00:00:35]Our goal is to find the length of segment X.

[00:00:38]Before we proceed, please take a moment to hit the like button and subscribe to my channel.

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[00:00:50]Also, feel free to pause the video and give it a try and tell us your answer in the comment section.

[00:00:57]To solve this problem, first, let's focus on the rectangle.

[00:01:02]Since PQRS is a rectangle, the opposite sides PQ and SR are parallel.

[00:01:12]By the alternate interior angles theorem, angle QPS is equal to angle PSR.

[00:01:30]We are given from the question that angle QPS is equal to angle TRP.

[00:01:41]Therefore, by transitivity, angle TRP is equal to angle PSR, which can also be named angle TSR.

[00:01:55]Now, let's consider two triangles TRP and TSR.

[00:02:04]Both triangles share the angle at vertex T.

[00:02:07]That is angle PTR is the same as angle RTS.

[00:02:17]Angle TRP is equal to angle TSR.

[00:02:25]By the angle-angle criterion, triangle TRP is similar to triangle TSR.

[00:02:34]And the proportionality of their corresponding sides is the ratio of TR to TS is equal to the ratio of TP to TR.

[00:02:50]By cross multiplication, the square of TR is equal to TP multiplied by TS.

[00:03:02]From the given information, TP is equal to 8.

[00:03:07]PS is equal to 10.

[00:03:12]TS is equal to TP plus PS.

[00:03:20]Which is equal to 8 plus 10, and 8 plus 10 is equal to 18.

[00:03:26]Substituting these values into the equation will give us the square of TR is equal to 8 multiplied by 18.

[00:03:37]8 multiplied by 18 is equal to 144.

[00:03:42]So, the square of TR is equal to 144.

[00:03:47]Taking the square root of both sides will give us TR is equal to 12.

[00:03:54]Therefore, the length of segment X is 12 units.

[00:03:59]Thanks for watching.

[00:04:00]Don't forget to like and subscribe for more videos.