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# Area With Determinants - Problem 1

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We’ve learned that we can use determinants to find the area of a parallelogram. But it turns out you can also use them to find the area of a triangle. Let’s take a look at a problem; let B equal the point (4, 3), A equal (7, -1), R equal (0, 2) and T equal (-3, 6). Now the quadrilateral BART happens to be a parallelogram. That’s going to be very helpful to us. Find the area of triangle BAT.

Now I’ve got B, A, R and T all plotted here. Here’s my parallelogram BART and here’s my triangle BAT. The thing we have to observe here, is that the area of triangle BAT is half the area of the parallelogram. So we know how to find the area of the parallelogram, we’ll just have to multiply that by ½. Let’s get started with that.

Let me write that down first. The area of triangle BAT is going to be ½ the area of parallelogram BART. Now if I wanted to find the area of this parallelogram, I’m going to need 2 vectors. Let’s say I focus on point B and the two vectors coming out of point B; vector BA and vector BT. So I’m going to need the components of those two vectors.

Now BA is the vector going from point B to point A, so it’s 7 minus 4, 3; -1 minus 3, -4. And BT is the vector going from B to point T, so this will be -3 minus 4, -7, and 6 minus 3, 3. So I have components for BA and BT.

Now remember the area formula for the parallelogram BART is going to be the absolute value, so ½, the absolute value, of the determinant followed by these two guys. So 3, -4, -7, 3. That’s ½ absolute value of, I have 9 minus 28. 9 minus 28 is -17. The absolute value of -17 is 17. I get ½ of 17 which is 8.5. So the area of triangle BAT is 8.5.

Remember we calculated here 17, that’s the area of the entire parallelogram. Whenever you draw a diagonal in a parallelogram, it divides the parallelogram into two triangles of equal area. So each of these little triangles is going to have an area of 8½.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

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## Adan Cisneros · 9 months ago

Hi, 9-28 is not -17. 9-28=-19 Therefore, area of parallelogram BART= 19 units^2, and area of triangle BAT= (1/2)(19)= 9.5 units^2.