Video Transcript
Find the area of the triangle ๐ด๐ต๐ถ with vertices ๐ด: one, four; ๐ต: negative four, five; and ๐ถ: negative four, negative five.
At this problem, we can actually see that we can actually use the determinant to help us find the area of the triangle ๐ด๐ต๐ถ. And in order to do that, weโre actually gonna use an equation. So we can say that the area of a triangle with coordinates ๐, ๐; ๐, ๐; ๐, ๐ is the absolute value of ๐ด given that ๐ด is equal to a half of the determinant of the matrix ๐, ๐, one; ๐, ๐, one; ๐, ๐, one. So great, now that we have a formula that we can use, we can actually put our values in and calculate ๐ด.
What Iโve done first of all is Iโve actually labelled our coordinates. And this is gonna help us cause it can help us see whatโs gonna go into our formula. So weโve got ๐, ๐, ๐, ๐, ๐, ๐. So we can say that ๐ด is equal to a half multiplied the determinant of the matrix. And the top row is gonna be one, four, one because thatโs our ๐, ๐ and then one. Our next row will be negative four, five, one. Again, cause now weโre moving on to ๐, ๐. And then finally, our bottom rowโs going to be negative four, negative five, one. And again now, weโve substituted in our values for ๐, ๐.
Excellent! So now letโs calculate what the value of ๐ด is. Before we can calculate what the value of the determinant is, we just remind ourselves that when weโre calculating the value of the determinant, we need the positive, negative, positive above our columns. And the reason that is cause it actually helps us decide whether our coefficient, so the top row number, is going to be positive or negative. Okay, great. So letโs get on with it.
So first of all, we know that our coefficient is gonna be one because thatโs our first term in the first row. Okay, now what we do is we actually eliminate the same row and column that oneโs in. And then weโre concerned with the submatrix of the bottom four numbers here, which Iโve put an orange box around. And we can just remind ourself that if we want to find the determinant of a-a submatrix, a two-by-two submatrix, thatโs actually gonna be equal to ๐๐ minus ๐๐. Okay, great. So we can use that and we can work out ours. So weโre gonna have five multiplied by one, cause thatโs like our ๐ and our ๐. And thatโs minus one multiplied by negative five. Okay, great. Weโve done that part. Let me move on to the next part.
So our next coefficient is gonna be negative four. And thatโs a four because itโs the second term in the first row. And itโs negative cause we know that the second column is going to be negative. So weโve got negative four. So now weโre gonna multiply that negative four by the determinant of the submatrix negative four, one, negative four, one. And itโs that because weโve actually deleted the values in the row that the fourโs in and the column that the fourโs in. So weโre gonna have negative four multiplied by one minus one multiplied by negative four. Okay, great.
And now we can move on to the final part where weโre gonna have positive one as a coefficient. And then inside, weโve got negative four multiplied by negative five. Thatโs cause thatโs like our ๐ and our ๐ in our submatrix. And thatโs minus five multiplied by negative four. Okay, great. And then finally, weโre gonna divide the whole thing by two because if we look back to the original equation, it was a half multiplied by the determinant of our matrix.
Okay, great. So now letโs work out our values. Okay, so this gives us that ๐ด equals one multiplied by 10 minus four multiplied by zero plus one multiplied by 40, all divided by two. So therefore, ๐ด is equal to 50 over two which is equal to 25.
So now, we just check back cause it says the area of a triangle with the coordinates ๐, ๐; ๐, ๐; ๐, ๐ is the absolute value of ๐ด. Well, we look down to our value of ๐ด. Itโs already positive 25. So we donโt have to worry about it cause in the absolute value weโre only looking for the nonnegative, or positive answers. So therefore, we can say that the area of triangle ๐ด๐ต๐ถ is equal to 25 square units.
