What is the ratio of the area of the smaller triangle in relation to the area of the larger triangle?

Question

big triangle; 5,9. 2,3. 5,3

small triangle: 12.6. 10,2. 12,2

Answer 1

To find the ratio of the areas of the two triangles, we need to calculate the area of each triangle first.

Using the coordinates given for the vertices of the large triangle, we can use the Shoelace Formula to find its area.

The coordinates for the large triangle are: (5, 9), (2, 3), and (5, 3).

Using the formula:
Area = (1/2) * |(5*3 + 2*3 + 5*9) - (9*2 + 3*5 + 3*5)|
= (1/2) * |(15 + 6 + 45) - (18 + 15 + 15)|
= (1/2) * |66 - 48|
= (1/2) * 18
= 9

Therefore, the area of the large triangle is 9.

Now let's calculate the area of the small triangle.

The coordinates for the small triangle are: (12, 6), (10, 2), and (12, 2).

Using the same formula as above:
Area = (1/2) * |(12*2 + 10*2 + 12*6) - (6*10 + 2*12 + 2*12)|
= (1/2) * |(24 + 20 + 72) - (60 + 24 + 24)|
= (1/2) * |116 - 108|
= (1/2) * 8
= 4

Therefore, the area of the small triangle is 4.

To find the ratio of the area of the smaller triangle to the area of the larger triangle, we divide the area of the small triangle by the area of the large triangle:

Ratio = Area of Small Triangle / Area of Large Triangle
= 4 / 9
= 4:9

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