BD is a perpendicular bisector of triangle ABC. XZ is a perpendicular bisector of triangle WXY.
Triangle ABC is similar to triangle WXY with a scale factor of 1:2.
Part I. Find BD, AB, WZ, XZ and WX.
Part II. What is the scale factor of the perimeters of triangle ABC and triangle WXY?
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Part I.
Since BD is the perpendicular bisector of triangle ABC, it divides the side AC into two equal parts. This means that BD is also equal to half of AC. To find the length of BD, you need to know the length of AC.
Similarly, XZ is the perpendicular bisector of triangle WXY, meaning it divides side WY into two equal parts. Therefore, XZ is also equal to half of WY. To find the length of XZ, you need to know the length of WY.
Triangle ABC is similar to triangle WXY with a scale factor of 1:2. This means that the corresponding sides of these two triangles are in the ratio of 1:2.
Let's assume that the length of AB is x. Since AB is part of the side AC in triangle ABC and is also a corresponding side to WY in triangle WXY, the length of WY would be 2x.
Now, we can find the lengths of BD, AB, WZ, XZ, and WX.
- BD: Since BD is equal to half of AC, and we know that AB is x, then AC would be 2x. Therefore, BD is equal to half of AC, which is (1/2) * 2x = x.
- AB: Given that AB is x.
- WZ: Since ABC is similar to WXY with a scale factor of 1:2, the corresponding side WZ would also be in the ratio of 1:2. Therefore, WZ is equal to half of WX, which is (1/2) * 2x = x.
- XZ: Given that XZ is equal to half of WY, and WY is 2x, then XZ is equal to (1/2) * 2x = x.
- WX: Given that WX is 2x.
Part II.
The scale factor of the perimeters is the ratio of the perimeters of the two triangles.
The perimeter of triangle ABC is the sum of the lengths of its sides, which are AB, BC, and AC. Since AB and AC are both x, and BC is the side opposite to AB, it is also x (due to symmetry in a perpendicular bisector). Therefore, the perimeter of triangle ABC is 3x.
Similarly, the perimeter of triangle WXY is the sum of the lengths of its sides, which are WX, XY, and WY. Since WX is 2x, XY is 2x (due to similarity with a scale factor of 1:2), and WY is 2x, the perimeter of triangle WXY is 6x.
Therefore, the scale factor of the perimeters of triangle ABC and triangle WXY is 3x:6x, which simplifies to 1:2.