Can you find area of the Purple shaded Triangle? | (Square) | #math #maths | #geometry

Added: 2026-05-10

[00:00:00]Welcome to premath. In this video, we have got this green shaded triangle DEB and this purple shaded triangle BDF are fully confined in a blue square ABCD as you can see in this given diagram such that the area of this green shaded triangle has been given to us as 60 cm squared and moreover this segment AE length is 7 cm whereas this segment BF length is 17 cm.

[00:00:39]And now our task is to calculate the area of this purple shaded triangle uh BDF.

[00:00:49]Please don't forget to give a thumbs up and subscribe. And please keep in mind that this figure may not be 100% true to the scale.

[00:00:58]Let's go ahead and get started and here's our very first step. Let's focus on this blue square ABCD.

[00:01:06]If I label this side length of this square as X, then this side length is going to be X X and X across the board. And now we know that this segment AE length is 7 cm. So therefore this remaining segment DE length has got to be X minus 7. And now we are going to focus on this green shaded triangle BDE.

[00:01:37]And we can see the area of this triangle has been given to us as 60. And now let's recall the area of a triangle formula. Area is always equal to a half times base times the height of the triangle. And here we can see the base of this green shaded triangle is X minus 7. The height of this green shaded triangle is x and the area has been given to us as 60.

[00:02:05]So, therefore, we can write our area 60 equals to 1/2 times uh x uh times uh x - uh 7. And now I'm going to multiply both sides by two to remove this fraction.

[00:02:25]And here we can see this 1 / 2 and 2 is gone. So, we are ended up with 120 equals to and we are now going to distribute. That is going to give us uh x² - 7x. And now I'm going to move this 120 on the other side. So, therefore, we can write x² - 7x uh - 120 is going to be equal to zero.

[00:02:53]So, thus we are ended up with this quadratic equation x² - 7x - 120 equal to zero and we are going to solve it by grouping and factoring.

[00:03:04]So, therefore, I'm going to tweak this middle term. -7x could be written as -15x + 8 * x. As you can see in this next step, I have replaced this -7x with -15x + 8x.

[00:03:23]And now we can see x is in common for first two terms and eight is common for last two terms. So, therefore, we can write x * x - 15 + 8 * x - 15 equals to zero. And here we can see x - 15 and x - 15, they are in common. So, therefore, our factors are going to be x-15 * x + 8 = 0. And now we are going to separate them. x - 15 = 0. And the other one is going to be x + 8 = 0. For this first equation, our x value is going to be positive 15. And for this second equation, x value is going to be negative 8.

[00:04:22]And now we can see x = -8 value is not feasible, since x represents the side length. So, therefore, we are going to reject this negative value of x, and we are going to accept x = 15.

[00:04:38]So, thus we can see our x value turns out to be 15. So, therefore, x value I'm going to replace with 15.

[00:04:46]15 over here, 15 over here, and likewise, 15 side length over here. And now we are going to focus on this right triangle BCF. And I'm going to label this segment CF length as lowercase a. And now our task is to find the value of lowercase a.

[00:05:09]So, therefore, we are going to apply the Pythagorean theorem on this triangle.

[00:05:14]And here's our Pythagorean theorem, a² + b² = c². And in our case, our hypotenuse is 17, whereas our two legs are lowercase a and 15.

[00:05:27]Let's go ahead and fill in the blanks in this Pythagorean formula. So, we got a² + 15² = 17².

[00:05:38]Let's simplify a² + 225 = 289.

[00:05:48]And now I am going to subtract 225 from both sides and here we can see this is gone.

[00:05:58]So therefore a square value is going to be equal to 64 and now I'm going to undo this square by taking a square root on both the sides.

[00:06:10]So therefore our lower case uh a value turns out to be positive 8 cm.

[00:06:19]So therefore our this lower case a value turns out to be 8.

[00:06:25]And now let's make an observation. We can see this whole side length is 15 cm and this segment CF length is 8.

[00:06:37]So therefore this remaining segment DF length has got to be 7 cm. And finally we are going to focus on this purple shaded triangle BDF and we are going to calculate the area of this purple shaded triangle.

[00:06:55]And let's recall the area of a triangle formula once again. Area equals to a half times base times the height of the triangle and in our case the base of this purple shaded triangle is 7 whereas the height of this purple shaded triangle is 15. So therefore purple shaded triangle area is going to be a half times the base is 7 and the height is uh 15.

[00:07:23]If we simplify that is going to give us 105 divided by 2 and that is going to be equal to 52.5 cm squared the area of this purple shaded triangle. So thus after all the calculations and manipulations the area of the purple shaded triangle turns out to be 52.5 cm squared. In other words, the area of this purple shaded triangle is going to be 52.5 cm squared. And that's our final answer.

[00:07:58]Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.