Triangles inside a rectangle

A B C D is a rectangle. A B is 18 centimetres. B C is 10 centimetres. E is a point on D C. Find the area of triangle A B E. Write the rule first. Area of a triangle is half of base times height. The base is the side A B. A B stays 18 centimetres. The height is the shortest distance from the tip down to the base. It meets A B at a right angle. That distance is 10 centimetres. Now slide E along the far side. The triangle changes shape. Watch the height travel with it. It never gets longer and it never gets shorter. So the area does not change. So put the two numbers into the rule. Half of 18 times 10. Cross out the 2 under the line, and the 18 becomes 9. 9 times 10 is 90. Triangle A B E has an area of 90 square centimetres. Now run the rule backwards. A B C D is a rectangle. A B is 16 centimetres. E is a point on D C. The area of triangle A B E is 96 square centimetres. Find the length of B C. Area of a triangle is half of base times height. The base is A B, and it is 16 centimetres. The height is the distance across to D C. That distance is the side B C, and B C is what we want. Half of 16 times B C is 96. Now go back the other way. Double the area, then divide by the base. 96 times 2, divided by 16, is 12. B C is 12 centimetres. Your turn. P Q R S is a square. Its diagonal P R is 14 centimetres. Find the area of the square P Q R S. Try it first. Pause if you like. P Q R S is standing on its corner. Draw the smallest upright square around it. Call that square the box. P R is one diagonal, and it is 14 centimetres. Slide it down to the foot of the box. It fits exactly, so the box is 14 across. S Q is the other diagonal, and it is 14 too. Slide it over to the side of the box. So the box is 14 by 14. The two diagonals cut the box into four small rectangles. Look at one of them. A side of P Q R S is its diagonal, so it cuts that rectangle in two. One of the two pieces is inside P Q R S. The same thing happens in all four rectangles. So P Q R S is half of the box. 14 times 14 is 196. The box has an area of 196 square centimetres. 196 divided by 2 is 98. P Q R S has an area of 98 square centimetres. The diagonal reaches corner to corner across the box. P Q R S is half of that box. The four corner pieces outside it are the other half. Half of base times height is half of the box.