In rectangle ACDE, AB=5cm, C=10cm, and BE=13cm (triangle within the rectange is BDE). What is the total area outside the triangle?

Assuming that (i) there is a point B between A and C along one side of the rectangle, and that (ii) you mean "BC=10cm" instead of "C=10cm", then it seems to me you have a right-angled triangle ABE with the hypoteneuse BE which has length 13cm, and length AB = 5cm, so the other side AE has length 12cm (=sqrt(13^2 - 5^2) by Pythagoras). B lies between A and C, and AB=5cm and BC=10cm, so the complete length is AC=15cm. The length of the other side of the rectangle (i.e. AE) is 12cm, so the area of the rectangle is 12cm x 15cm = 180cm^2. The area of the right-angled triangle is half the base times the height, which is 5cm * 12cm / 2 = 30cm^2. So the total area outside the triangle is (180-30) = 150cm^2.

To find the area outside the triangle in rectangle ACDE, we need to subtract the area of triangle BDE from the total area of the rectangle.

To calculate the area of triangle BDE, we can use the formula: A = (1/2) * base * height.

The base of triangle BDE is BE, which is given as 13cm. However, we need to find the height of the triangle.

Looking at the rectangle, AC is a vertical side and BE is the diagonal. Since the opposite sides of a rectangle are equal, CD must also be 10cm. Therefore, we can consider triangle BCD as a right triangle.

By using the Pythagorean theorem, we can find the height of triangle BCD and use it as the height of triangle BDE.

The Pythagorean theorem states that the square of the hypotenuse (BE) is equal to the sum of the squares of the other two sides (BC and CD).

Applying the theorem to triangle BCD, we have: BE^2 = BC^2 + CD^2
13^2 = BC^2 + 10^2
169 = BC^2 + 100
BC^2 = 169 - 100
BC^2 = 69
BC = √69, approximately 8.31cm (rounded to two decimal places)

Now that we have the base (BE = 13cm) and height (BC = 8.31cm) of triangle BDE, we can calculate its area.

Area of triangle BDE = (1/2) * base * height
= (1/2) * 13cm * 8.31cm
≈ 54.14 cm^2 (rounded to two decimal places)

The total area outside the triangle can be found by subtracting the area of triangle BDE from the total area of the rectangle.

The total area of the rectangle ACDE is given by: Area = length * width
In this case, length AC = 10cm and width AD = 5cm.

Area of rectangle ACDE = length * width
= 10cm * 5cm
= 50 cm^2

Total area outside the triangle = Area of rectangle - Area of triangle BDE
= 50 cm^2 - 54.14 cm^2
≈ -4.14 cm^2 (rounded to two decimal places)

Since the result is negative, it means that the triangle BDE is larger than the rectangle ACDE, which is not possible. Please recheck the values given for AB, C, and BE, or the dimensions of the rectangle and triangle.