A right triangle is dilated by a factor of 3.5.
Which statements about the resulting triangle must be true?
Select all that apply.
(There is more than one answer)
It is smaller than the original triangle.
It is an isosceles triangle.
It is larger than the original triangle.
It is a right triangle.
To determine which statements about the resulting triangle are true after a dilation, let's understand the properties of dilation and its effect on the triangle.
Dilation is a transformation in which every point of the original figure is multiplied by a common scale factor. In this case, the right triangle is dilated by a factor of 3.5, which means that each side of the original triangle is multiplied by 3.5.
1. It is smaller than the original triangle:
Since the dilation factor is greater than 1 (3.5), the resulting triangle will be larger than the original triangle, not smaller. Therefore, this statement is NOT true.
2. It is an isosceles triangle:
An isosceles triangle has at least two sides of equal length. However, dilation does not change the relative lengths of the sides of the triangle. Therefore, if the original triangle was not isosceles, then the resulting triangle won't be either. So, this statement is NOT necessarily true.
3. It is larger than the original triangle:
As mentioned earlier, a dilation factor of 3.5 means that each side of the original triangle is multiplied by 3.5. This multiplication effectively increases the size of each side by a factor of 3.5. Thus, the resulting triangle will be larger than the original triangle. Therefore, this statement is true.
4. It is a right triangle:
A right triangle is a triangle that contains a 90-degree angle. Dilation does not change the angles of a triangle, only the lengths of its sides. Therefore, if the original triangle was a right triangle, then the resulting triangle will also be a right triangle. So, this statement is true.
Based on the above explanation, the statements that must be true are:
- It is larger than the original triangle.
- It is a right triangle.
The correct statements about the resulting triangle after dilating a right triangle by a factor of 3.5 are:
1. It is larger than the original triangle.
2. It is a right triangle.
Explanation:
1. When a figure is dilated by a scale factor larger than 1, such as 3.5, it increases in size. Therefore, the resulting triangle after dilation is larger than the original triangle.
2. Dilating a right triangle by any scale factor preserves the angles, including the right angle. Therefore, the resulting triangle is still a right triangle.
The options "It is smaller than the original triangle" and "It is an isosceles triangle" are not necessarily true when a right triangle is dilated by a factor of 3.5.
It is clearly larger, and it remains its relative shape, so
it is still a right-angled triangle