Question

In the picture, lines a, b, and c are parallel. Simon and Adara set up correct but different proportions that can be used to find the unknown segment lengths in the diagram. Use the drop-down menus to complete the proportions.

Answer 1

a/b = c/d; for parallel lines a, b, and c, this proportion relates the corresponding segments in similar triangles formed by these lines.

Step-by-step explanation:

In the given diagram where lines a, b, and c are parallel, we can observe that triangles formed by these lines are similar due to the corresponding angles being congruent. When dealing with similar triangles, the ratio of corresponding sides remains constant. Using this property, the proportions to find the unknown segment lengths can be set up. One correct proportion that can be formed is a/b = c/d.

Consider the triangles formed by the parallel lines. By identifying the corresponding sides of these similar triangles, we establish the proportion a/b = c/d. This relationship arises from the fact that in similar triangles, corresponding sides are in proportion to each other. Therefore, the ratio of the length of side a to side b is equal to the ratio of the length of side c to side d.

To further illustrate this, it's important to note that the lines being parallel ensure that the angles between them are congruent, leading to the formation of similar triangles. Consequently, the proportions set up using the corresponding sides of these similar triangles hold true and can be used to solve for unknown segment lengths when appropriate values are given for other sides.

Answer 2

The proportions to find the unknown segment lengths in the diagram should be completed as follows;

Simon's Proportion
`(x+4)/(7x-4) =(3)/(5)`

Adara's Proportion
`(3)/(8) =(x+4)/(8x)`.

In Mathematics and Euclidean Geometry, the basic proportionality theorem states that when any of the two sides of a triangle is intersected by a straight line which is parallel to the third side of the triangle, then, the two sides that are intersected would be divided proportionally and in constant ratio.

By applying the basic proportionality theorem to the diagram, we have the following proportional side lengths as Simon's equation:

`(x+4)/(7x-4) =(3)/(5)`

For Adara's Proportion, we have the following equation;

`(3)/(8) =(x+4)/(x+4+7x-4)\\\\(3)/(8) =(x+4)/(8x)`

Missing information;

In the picture, lines α, b, and c are parallel. Simon and Adara set up correct but different proportions that can be used to find the unknown segment lengths in the diagram. Use the drop-down menus to complete the proportions.