The base (BC) of an isosceles triangle (ABC) is also the side of a square (BCEF) in which the triangle is inscribed. A perpendicular line is drawn in the triangle from B to AC at D, forming triangle ABD. What is the ratio of triangle ABD to the square BCEF?
a graphical solution seems the most straight forward
B(0,0)
C(10,0)
E(10,10)
A(5,10)
F(0,10)
find the equation for AC, and then the equation for BD
D is the intersection of the lines
calculate the lengths of BD and AD to find the area of ABD
Assuming you mean the ratio of areas,
If BC = 2x, then AD = 2x
ABD has area 1/2 * 2x * 2x = 2x^2
ABEF has area (2x)*(2x) = 4x^2
AC ... y = -2 x + 20
BD ... y = 1/2 x
D (8,4)
BD = √(64 + 16)
AD = √(9 + 36)
area ABD = ½ √(80 * 45) = 30
area ABEF = 10² = 100
To find the ratio of triangle ABD to the square BCEF, we need to compare their areas.
First, let's consider triangle ABD. Since triangle ABC is an isosceles triangle, we know that BD is the perpendicular bisector of AC. This means that AD and DC have the same length, making triangle ABD a right triangle.
Since BD is the base of the square BCEF, and triangle ABD is a right triangle with BD as the hypotenuse, we can conclude that the length of BD is the same as the side length of the square, let's say s.
Now, let's calculate the area of triangle ABD. The area of a right triangle can be calculated using the formula: area = (base * height) / 2. In this case, the base is AD and the height is BD.
Since AD is half of AC (since it's the perpendicular bisector), and AC is the base of the isosceles triangle ABC, we can say that AD = BC/2.
Thus, the area of triangle ABD can be calculated as: area_ABD = (AD * BD) / 2 = (BC/2 * s) / 2 = BC * s / 4.
Now, let's calculate the area of the square BCEF. The area of a square is calculated by squaring its side length. In this case, the side length of the square is s.
So, the area of square BCEF is: area_BCEF = s^2.
Finally, to find the ratio of triangle ABD to square BCEF, we divide the area of triangle ABD by the area of square BCEF:
ratio = area_ABD / area_BCEF = (BC * s / 4) / (s^2) = BC / (4s).
Therefore, the ratio of triangle ABD to the square BCEF is BC / (4s).