Two triangles are similar, and the ratio of each pair of corresponding sides is 2 : 1. Which statement regarding the two triangles is not true?
1) Their areas have a ratio of 4 : 1.
2) Their altitudes have a ratio of 2 : 1.
3) Their perimeters have a ratio of 2 : 1.
4) Their corresponding angles have a ratio of 2 : 1.
Look at answer 4).
Can two similar triangles have a ratio of two for the corresponding angles?
What happens to the sum of the angles?
Two similar polygons have areas of 4 square inches and 64 square inches.
To determine which statement regarding the two triangles is not true, we need to consider the properties of similar triangles.
When two triangles are similar, it means that their corresponding angles are equal and the ratio of their corresponding sides is constant. In this case, the ratio of each pair of corresponding sides is given as 2:1.
1) The statement that their areas have a ratio of 4:1 is true. The ratio of the corresponding sides is 2:1, which means the ratio of their areas is equal to the square of the corresponding sides' ratio. The square of 2:1 is 4:1, so their areas have a ratio of 4:1. This statement is true.
2) The statement that their altitudes have a ratio of 2:1 is also true. Since the triangles are similar, the ratio of their corresponding sides is 2:1. The altitudes of triangles are perpendicular segments drawn from a vertex to the opposite side. The ratio of their altitudes is the same as the ratio of their corresponding sides. Therefore, this statement is also true.
3) The statement that their perimeters have a ratio of 2:1 is true. Since the ratio of each pair of corresponding sides is 2:1, it means that the lengths of the corresponding sides are in the ratio 2:1. The perimeter of a triangle is the sum of the lengths of its sides. So, if the corresponding sides have a ratio of 2:1, their perimeters will also have a ratio of 2:1. Hence, this statement is true.
4) The statement that their corresponding angles have a ratio of 2:1 is not true. While the two triangles are similar, the corresponding angles are actually equal, not in a specific ratio. So, this statement is not true.
Therefore, the statement that is not true is option 4) Their corresponding angles have a ratio of 2:1.