(5,9) (2,3) (5,3)

Question

(12,6) (10,2) (12,2)
what is the ratio of the area of the smaller triangle in relation to the area of the larger triangle?

Answer 1

To find the ratio of the area of the smaller triangle to the larger triangle, we can find the area of each triangle and then divide.

Let's label the points of the triangles: A, B, and C for the larger triangle, and D, E, and F for the smaller triangle.

For the larger triangle with points A(5, 9), B(2, 3), and C(5, 3):

Let's find the base of the larger triangle AC:

Distance AC = √((5-5)^2 + (3-9)^2) = √(0^2 + 6^2) = √36 = 6.

The height of the larger triangle is the vertical distance from point B(2, 3) to the line AC, which is parallel to the y-axis as the x-coordinates of points A and C are the same.

Therefore, the height of the larger triangle is: height = 9 - 3 = 6.

The area of the larger triangle ABC is given by: 1/2 * base * height = 1/2 * 6 * 6 = 18 square units.

For the smaller triangle with points D(12, 6), E(10, 2), and F(12, 2):

Let's find the base of the smaller triangle DF:

Distance DF = √((12-12)^2 + (2-6)^2) = √(0^2 + (-4)^2) = √16 = 4.

The height of the smaller triangle is the vertical distance from point E(10, 2) to the line DF, which is parallel to the y-axis as the x-coordinates of points D and F are the same.

Therefore, the height of the smaller triangle is: height = 6 - 2 = 4.

The area of the smaller triangle DEF is given by: 1/2 * base * height = 1/2 * 4 * 4 = 8 square units.

The ratio of the area of the smaller triangle DEF to the larger triangle ABC is: 8/18 = 4/9.

So, the ratio of the area of the smaller triangle to the larger triangle is 4/9.