Given trapezoid WXYZ, find the ratio of the areas of each pair of triangles:
WYX and XYZ
WXZ and WXY
To find the ratio of the areas of each pair of triangles in the trapezoid WXYZ, let's consider the sides and bases of the trapezoid.
1. Ratio of areas for triangles WYX and XYZ:
The triangles WYX and XYZ share the same base, YX. To find the ratio of their areas, we need to compare their heights.
Let h₁ be the height of triangle WYX, and h₂ be the height of triangle XYZ.
Since the trapezoid WXYZ is a right trapezoid, the height of the trapezoid is the perpendicular distance between the bases WX and YZ.
Therefore, the ratio of the heights h₁ and h₂ is equal to the ratio of the areas of triangles WYX and XYZ:
Ratio of areas = h₁ / h₂.
2. Ratio of areas for triangles WXZ and WXY:
The triangles WXZ and WXY also share the same base, WX. To find the ratio of their areas, we need to compare their heights.
Let h₃ be the height of triangle WXZ, and h₄ be the height of triangle WXY.
Again, since the trapezoid WXYZ is a right trapezoid, the height of the trapezoid is the perpendicular distance between the bases WX and YZ.
Therefore, the ratio of the heights h₃ and h₄ is equal to the ratio of the areas of triangles WXZ and WXY:
Ratio of areas = h₃ / h₄.
So, to find the ratio of the areas for each pair of triangles, you need to determine the heights (perpendicular distances) of the triangles involved.
To find the ratio of the areas of each pair of triangles in trapezoid WXYZ, we need to calculate the areas of the individual triangles and then compare them.
Let's start by labeling the trapezoid:
W is the leftmost vertex,
X is the rightmost vertex,
Y is the upper left vertex,
Z is the upper right vertex.
1) Ratio of the areas of triangles WYX and XYZ:
To calculate the area of triangle WYX, we need the length of the base (WX) and the height of the triangle (the perpendicular distance between the base WX and the vertex Y). Similarly, to calculate the area of triangle XYZ, we need the length of the base (WZ or XZ) and the height of the triangle (the perpendicular distance between the base WZ or XZ and the vertex Y).
Once we have the lengths of the bases and the heights, we can calculate the areas using the formula: Area = (base * height) / 2.
2) Ratio of the areas of triangles WXZ and WXY:
To calculate the area of triangle WXZ, we need the length of the base (WX or XZ) and the height of the triangle (the perpendicular distance between the base WX or XZ and the vertex Y or Z). Similarly, to calculate the area of triangle WXY, we need the length of the base (WY) and the height of the triangle (the perpendicular distance between the base WY and the vertex X or Y).
Once we have the lengths of the bases and the heights, we can calculate the areas using the formula: Area = (base * height) / 2.
Finally, to find the ratio of the areas, divide the area of the first triangle by the area of the second triangle.
By following these steps, you can find the ratio of the areas for each pair of triangles in trapezoid WXYZ.
I am saying xy and wz are parallel (does not matter but choosing that)
defining:
wz = a
xy = b
height common is h
then
Awyx = .5 h b
Axyz = .5 h a
ratio = b/a
Awxz = .5 h a
Awxy = .5 h b
ratio = a/b