Determine whether the polygons are similar.
ABCD with vertices A(–5, 4), B(–2, 4), C(–2, 2), D(–5, 2)
and EFGH with vertices E(–2, 0), F(4, 0), G(4, –6), H(–2, –6)
To be similar, corresponding sides must be in the same ratio
that is AB/EF = BC/FG = etc
Find the lengths and test if the ratios are the same
They're not similar according to my calculations. Is this right?
determine whether the polygons in each pair are similar
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To determine whether the polygons ABCD and EFGH are similar, we need to check if their corresponding angles are congruent and their corresponding sides are proportional.
1. Corresponding angles: Compare the measures of the corresponding angles of the two polygons. Compute the angle measures formed by the sides of each polygon. We can use the slope formula to find the angle measure between two line segments:
a) For ABCD:
- Angle A = slope of line segment AB - slope of line segment AD
- Angle B = slope of line segment BC - slope of line segment BA
- Angle C = slope of line segment CD - slope of line segment CB
- Angle D = slope of line segment DA - slope of line segment DC
b) For EFGH:
- Angle E = slope of line segment EF - slope of line segment EH
- Angle F = slope of line segment FG - slope of line segment FE
- Angle G = slope of line segment GH - slope of line segment GF
- Angle H = slope of line segment HE - slope of line segment HG
If each corresponding pair of angles (i.e., Angle A and Angle E, Angle B and Angle F, Angle C and Angle G, Angle D and Angle H) have the same measure, then the polygons ABCD and EFGH have congruent corresponding angles.
2. Corresponding sides: Compute the lengths of the corresponding sides of each polygon. Use the distance formula to find the distance between two points:
a) For ABCD:
- Side AB = distance between points A and B
- Side BC = distance between points B and C
- Side CD = distance between points C and D
- Side DA = distance between points D and A
b) For EFGH:
- Side EF = distance between points E and F
- Side FG = distance between points F and G
- Side GH = distance between points G and H
- Side HE = distance between points H and E
If the ratios of the corresponding side lengths (AB/EF, BC/FG, CD/GH, DA/HE) are all equal, then the polygons ABCD and EFGH have proportional corresponding sides.
If both the corresponding angles and sides are congruent and proportional respectively, then the polygons ABCD and EFGH are similar.