Units on the figure

In triangle A B C, D is a point on B C. The ratio of B D to D C is 2 to 3. The area of triangle A B C is 60 square centimetres. Find the area of triangle A B D. Both small triangles have the same tip A. Their bases lie on the same line. So the distance from A down to that line is the same for both. Each area is half of its own base, times that same height. The half is the same in both. The height is the same in both. Only the base is different. So the areas follow the bases. So mark the base in units before any area. B D to D C is 2 to 3. That is not two thirds. B D is 2 parts. D C is 3 parts. The whole base is 5 parts. 5 units of area is 60 square centimetres. So 1 unit of area is 12 square centimetres. Triangle A B D is 2 units. Its area is 24 square centimetres.

Now use units inside a rectangle. A B C D is a rectangle. E is a point on A B. A E is 3 times as long as E B. The area of triangle A E D is 36 square centimetres. Find the area of the rectangle A B C D. Mark A E as 3 units. Mark E B as 1 unit. A B is 4 units altogether. Now look at triangle A E D. Its base is A E, and A E is 3 units. Its height is A D, the side of the rectangle. So the area of A E D is half of 3 units, times A D. And the question tells us that is 36. Undo the half. Multiply both sides by 2. So 3 units times A D is 72. That is this rectangle, 3 units wide. Share it between the 3 units. 1 unit times A D is 72 divided by 3. That is 24. Here is 1 unit times A D on the figure. The whole rectangle is 4 of them. 4 times 24 is 96 square centimetres. The question asked for the area of A B C D. That last line is the one that answers it. And this is the area of A B C D.

Your turn. The figure is made up of a square A B C D and a triangle C D E joined along D C. The angle at D is a right angle. D E is twice as long as D C. The area of the whole figure is five hundred and twelve square centimetres. Find the length of D C. Try it first. Pause if you like. Look at the square first. Its area is D C times D C. Now look at the triangle. Its area is half of D C, times D E. And D E is 2 times D C. So write the line again, with 2 D C in place of D E. Now cancel the 2 underneath with the 2 on top. That leaves D C times D C. So the square and the triangle have equal areas. The area of the square plus the area of C D E is five hundred and twelve square centimetres. The two areas are the same, so each one is D C times D C. Two of the same thing, so that is 2 times D C times D C. So D C times D C is five hundred and twelve divided by 2. Two hundred and fifty six. The square root of two hundred and fifty six is 16. So D C is 16 centimetres. The marks say the height is two of the square's sides. So the two areas are equal. D C times D C is an area. The question asked for a length. The last line is the one that answers it.