Are quadrilaterals ABCD and EFGH similar?

Steps to Solve

1. Find the lengths of the sides of quadrilateral ABCD

Calculate the lengths of each side using the distance formula between the coordinates of points A, B, C, and D.

The distance formula is given by: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Calculate the lengths of the sides for ABCD

For sides AB, BC, CD, and DA, use the coordinates:

• Length AB: $d_{AB} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$
• Length BC: $d_{BC} = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2}$
• Length CD: $d_{CD} = \sqrt{(x_D - x_C)^2 + (y_D - y_C)^2}$
• Length DA: $d_{DA} = \sqrt{(x_A - x_D)^2 + (y_A - y_D)^2}$

3. Find the lengths of the sides of quadrilateral EFGH

Repeat the distance calculation for quadrilateral EFGH using its coordinates.

• Length EF: $d_{EF} = \sqrt{(x_F - x_E)^2 + (y_F - y_E)^2}$
• Length FG: $d_{FG} = \sqrt{(x_G - x_F)^2 + (y_G - y_F)^2}$
• Length GH: $d_{GH} = \sqrt{(x_H - x_G)^2 + (y_H - y_G)^2}$
• Length HE: $d_{HE} = \sqrt{(x_E - x_H)^2 + (y_E - y_H)^2}$

4. Compare the corresponding sides

Determine the ratios of the corresponding sides of quadrilaterals ABCD and EFGH. Check if these ratios are equal.

5. Check the angles (if necessary)

Using the dot product of the vectors corresponding to the sides (if necessary), verify that the angles between corresponding sides are the same.

The dot product method involves the formula: $$ \text{dot product} = x_1x_2 + y_1y_2 $$

6. Determine similarity

Based on the side ratios and angle equality, conclude if quadrilaterals ABCD and EFGH are similar.