Question
Parallelogram A B C D is shown. Line segment X Y goes from point X on side A B to point Y on side C D to form 2 trapezoids.
Figure ABCD is a parallelogram. Two trapezoids are created using line segment XY such that AX ≅ CY.
What is true about the areas of the trapezoids?
Each area is equal to half of the area of ABCD.
The area of AXYD is less than the area of BXYC.
The area of AXYD is greater than the area of BXYC.
Each area is equal to the area of ABCD.
Figure ABCD is a parallelogram. Two trapezoids are created using line segment XY such that AX ≅ CY.
What is true about the areas of the trapezoids?
Each area is equal to half of the area of ABCD.
The area of AXYD is less than the area of BXYC.
The area of AXYD is greater than the area of BXYC.
Each area is equal to the area of ABCD.
Answers
Answered by
GPT-4o mini
To analyze the areas of the trapezoids created by the line segment XY in parallelogram ABCD, we start with the knowledge that AX ≅ CY.
In this case, line segment XY divides parallelogram ABCD into two trapezoids: AXYD (the trapezoid formed by vertices A, X, Y, and D) and BXYC (the trapezoid formed by vertices B, X, Y, and C).
Since we are given that AX is congruent to CY, this means that the lengths from vertex A to point X and from vertex C to point Y are equal.
1. **Area of Parallelogram ABCD**: The area of the entire parallelogram can be expressed in terms of base and height. In this case, the height can be considered as the same for both trapezoids since XY is parallel to the bases AD and BC.
2. **Ratio of Areas of Trapezoids**: Since AX ≅ CY, the line segment XY effectively slices the parallelogram such that the triangles AXC and BYD (which are the triangles above and below line XY) will have equal heights but differing bases. However, these bases depend on the distances along the sides of the trapezoids.
Given that AX ≅ CY, and that the two trapezoids share the same height (from XY to line AD and line BC), it will mean that trapezoids AXYD and BXYC will cover equal width along the respective triangles they form. Thus, each trapezoid has a height that intercepts with equal segments of the parallelogram heights and bases because the cut regards equal segmentations of opposite sides.
Therefore, we can conclude that the areas of trapezoids AXYD and BXYC will each be half of the area of parallelogram ABCD.
Thus, the correct statement about the areas of the trapezoids is:
**Each area is equal to half of the area of ABCD.**
In this case, line segment XY divides parallelogram ABCD into two trapezoids: AXYD (the trapezoid formed by vertices A, X, Y, and D) and BXYC (the trapezoid formed by vertices B, X, Y, and C).
Since we are given that AX is congruent to CY, this means that the lengths from vertex A to point X and from vertex C to point Y are equal.
1. **Area of Parallelogram ABCD**: The area of the entire parallelogram can be expressed in terms of base and height. In this case, the height can be considered as the same for both trapezoids since XY is parallel to the bases AD and BC.
2. **Ratio of Areas of Trapezoids**: Since AX ≅ CY, the line segment XY effectively slices the parallelogram such that the triangles AXC and BYD (which are the triangles above and below line XY) will have equal heights but differing bases. However, these bases depend on the distances along the sides of the trapezoids.
Given that AX ≅ CY, and that the two trapezoids share the same height (from XY to line AD and line BC), it will mean that trapezoids AXYD and BXYC will cover equal width along the respective triangles they form. Thus, each trapezoid has a height that intercepts with equal segments of the parallelogram heights and bases because the cut regards equal segmentations of opposite sides.
Therefore, we can conclude that the areas of trapezoids AXYD and BXYC will each be half of the area of parallelogram ABCD.
Thus, the correct statement about the areas of the trapezoids is:
**Each area is equal to half of the area of ABCD.**