Video: Finding the Area of a Triangle given Its Vertices

Find the area of the triangle ๐ด๐ต๐ถ with vertices ๐ด (1, 4), ๐ต (โˆ’4, 5), and ๐ถ (โˆ’4, โˆ’5).

04:09

Video Transcript

Find the area of the triangle ๐ด๐ต๐ถ with vertices ๐ด: one, four; ๐ต: negative four, five; and ๐ถ: 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 ๐ด๐ต๐ถ. And in order to do that, weโ€™re actually gonna use an equation. So we can say that the area of a triangle with coordinates ๐‘Ž, ๐‘; ๐‘, ๐‘‘; ๐‘’, ๐‘“ is the absolute value of ๐ด given that ๐ด is equal to a half of the determinant of the matrix ๐‘Ž, 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 ๐ด.

What Iโ€™ve done first of all is Iโ€™ve actually labelled our coordinates. And this is gonna help us cause it can help us see whatโ€™s gonna go into our formula. So weโ€™ve got ๐‘Ž, ๐‘, ๐‘, ๐‘‘, ๐‘’, ๐‘“. So we can say that ๐ด is equal to a half multiplied the determinant of the matrix. And the top row is gonna be one, four, one because thatโ€™s our ๐‘Ž, ๐‘ and then one. Our next row will be negative four, five, one. Again, cause now weโ€™re moving on to ๐‘, ๐‘‘. And then finally, our bottom rowโ€™s going to be negative four, negative five, one. And again now, weโ€™ve substituted in our values for ๐‘’, ๐‘“.

Excellent! So now letโ€™s calculate what the value of ๐ด is. Before we can calculate what the value of the determinant is, we just remind ourselves that when weโ€™re 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 letโ€™s get on with it.

So first of all, we know that our coefficient is gonna be one because thatโ€™s our first term in the first row. Okay, now what we do is we actually eliminate the same row and column that oneโ€™s in. And then weโ€™re concerned with the submatrix of the bottom four numbers here, which Iโ€™ve 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, thatโ€™s actually gonna be equal to ๐‘Ž๐‘‘ minus ๐‘๐‘. Okay, great. So we can use that and we can work out ours. So weโ€™re gonna have five multiplied by one, cause thatโ€™s like our ๐‘Ž and our ๐‘‘. And thatโ€™s minus one multiplied by negative five. Okay, great. Weโ€™ve done that part. Let me move on to the next part.

So our next coefficient is gonna be negative four. And thatโ€™s a four because itโ€™s the second term in the first row. And itโ€™s negative cause we know that the second column is going to be negative. So weโ€™ve got negative four. So now weโ€™re gonna multiply that negative four by the determinant of the submatrix negative four, one, negative four, one. And itโ€™s that because weโ€™ve actually deleted the values in the row that the fourโ€™s in and the column that the fourโ€™s in. So weโ€™re 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 weโ€™re gonna have positive one as a coefficient. And then inside, weโ€™ve got negative four multiplied by negative five. Thatโ€™s cause thatโ€™s like our ๐‘Ž and our ๐‘‘ in our submatrix. And thatโ€™s minus five multiplied by negative four. Okay, great. And then finally, weโ€™re 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 letโ€™s work out our values. Okay, so this gives us that ๐ด equals one multiplied by 10 minus four multiplied by zero plus one multiplied by 40, all divided by two. So therefore, ๐ด 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‘Ž, ๐‘; ๐‘, ๐‘‘; ๐‘’, ๐‘“ is the absolute value of ๐ด. Well, we look down to our value of ๐ด. Itโ€™s already positive 25. So we donโ€™t have to worry about it cause in the absolute value weโ€™re only looking for the nonnegative, or positive answers. So therefore, we can say that the area of triangle ๐ด๐ต๐ถ is equal to 25 square units.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.