Can You find The Area of the Triangle? | (Triangle) #geometry

Added: 2026-05-10

[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 this question, we are given a large triangle ABC.

[00:00:13]In the triangle, a vertical line DC is drawn such that it is perpendicular to BC at point C.

[00:00:21]The length of BC is given as 10 units and the length of AC is given as 13 units.

[00:00:27]It is also given that BD is equal to twice the length of segment AD.

[00:00:33]Our goal is to find the area of triangle ADC.

[00:00:37]Before we proceed, please take a moment to hit the like button and subscribe to my channel.

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[00:00:49]Also, feel free to pause the video and give it a try and tell us your answer in the comment section.

[00:00:56]To solve this problem, first, let the length of segment AD be X.

[00:01:03]Since we are given that BD is equal to twice the length of AD, the length of BD will be 2X.

[00:01:11]Next, let's extend A and D until they intersect.

[00:01:17]Observe that the two lines are perpendicular to each other at the point of intersection.

[00:01:23]Let this point be E.

[00:01:27]Next, let's focus on triangle AED.

[00:01:31]Triangle AED is a right triangle.

[00:01:34]If we call this angle alpha, then the remaining angle of this triangle will be 90° minus alpha.

[00:01:46]Reason: The sum of angles in a triangle is 180°.

[00:01:52]Next, let's consider triangle BCD.

[00:01:55]Triangle BCD is also a right triangle.

[00:02:00]In this triangle, notice that angle CDB is alpha.

[00:02:05]Reason: vertically opposite angles are equal.

[00:02:10]So, the remaining angle of this triangle will be 90° - alpha.

[00:02:17]Reason: The sum of angles in a triangle is 180°.

[00:02:23]Comparing triangle BCD and triangle AED, observe that angle DBC is equal to angle DAE, angle BCD is equal to angle AED, and angle CDB is equal to angle EDA.

[00:03:12]This shows that triangle BCD is similar to triangle AED.

[00:03:25]And the proportionality of their corresponding sides is BC over AE is equal to BD over AD, which is equal to CD over DE.

[00:03:46]BC is equal to 10.

[00:03:51]BD is equal to 2 x and AD is equal to x.

[00:04:04]Substituting these values in this expression will give us 10 over AE is equal to 2 x over x.

[00:04:18]X will cancel out x.

[00:04:20]And by cross multiplication AE will be equal to 10 over 2.

[00:04:31]10 divided by 2 is 5.

[00:04:34]AE is equal to 5.

[00:04:41]Again, if we plug in the value of each segments in this expression, we will have 2 x over x is equal to CD over DE.

[00:04:57]X will cancel out x.

[00:05:00]And by cross multiplication CD will be equal to 2 * DE.

[00:05:09]So, if we take DE as y CD will be equal to 2 y.

[00:05:25]Going further let's focus on triangle ACE.

[00:05:29]Triangle ACE is a right triangle.

[00:05:36]From Pythagoras theorem AC squared is equal to AE squared plus CE squared.

[00:05:48]AC is equal to 13.

[00:05:52]And AE is equal to five.

[00:05:59]Substituting these values in the above expression will give us 13 squared is equal to five squared plus CE squared.

[00:06:13]13 squared is equal to 169.

[00:06:17]And five squared is equal to 25.

[00:06:25]Rearranging this will give us CE squared is equal to 169 minus 25.

[00:06:36]169 minus 25 is equal to 144.

[00:06:44]So, CE squared is equal to 144.

[00:06:49]Taking the square root of both sides will give us CE is equal to 12.

[00:07:01]Now, notice that Y plus 2Y is equal to 12.

[00:07:16]If we add Y and 2Y together we will have 3Y is equal to 12.

[00:07:26]Dividing through by three three will cancel out three.

[00:07:34]And 12 divided by three is four.

[00:07:37]And we are left with Y is equal to 4.

[00:07:46]Since DE is y and CD is 2y, it follows that DE is equal to 4.

[00:07:57]And CD is equal to 2 * 4, which is equal to 8.

[00:08:06]Now, to find the area of triangle ADC, let's recall that the area of triangle is 1/2 * base * height.

[00:08:18]So, the area of triangle ADC is equal to 1/2 * CD * AE.

[00:08:34]CD is equal to 8.

[00:08:39]And AE is equal to 5.

[00:08:45]If we plug in these values in the above expression, we will have 1/2 * 8 * 5.

[00:08:56]8 / 2 is 4.

[00:09:00]And we are left with 1 * 4 * 5, which is equal to 20.

[00:09:10]Hence, the area of triangle ADC is equal to 20 square units.

[00:09:16]Thanks for watching.

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