Question

The isosceles trapezoids, ABCD and EFGH, are similar quadrilaterals. The scale factor between the trapezoids is 2:3. Given that GH=6 centimeters, AD=8 centimeters, and AB is three times the length of DC:

Write a similarity statement for each of the four pairs of corresponding sides.
Write a congruence statement for each of the four pairs of corresponding angles.
Determine the perimeter for each of the isosceles trapezoids.
Use complete sentences to answer the following prompts and include all of your calculations in your final answer.

Answer 1

First I name the correspondence implied by the given similarity: ABCD ~ EFGH, so A ↔ E, B ↔ F, C ↔ G, and D ↔ H.

Similarity statements for corresponding sides (scale factor ABCD : EFGH = 2:3):
- AB : EF = 2 : 3
- BC : FG = 2 : 3
- CD : GH = 2 : 3
- DA : HE = 2 : 3

Congruence statements for corresponding angles:
- ∠A ≅ ∠E
- ∠B ≅ ∠F
- ∠C ≅ ∠G
- ∠D ≅ ∠H

Now find the side lengths and perimeters.

1) Use the scale factor with GH = 6 cm. Since CD corresponds to GH and ABCD is the smaller (2:3),
CD = (2/3) · GH = (2/3) · 6 = 4 cm.

2) AB is three times DC, so
AB = 3 · CD = 3 · 4 = 12 cm.

3) In an isosceles trapezoid the legs are equal, so BC = AD. Given AD = 8 cm,
BC = 8 cm.

Check sides of ABCD: AB = 12 cm, BC = 8 cm, CD = 4 cm, DA = 8 cm.
Perimeter(ABCD) = 12 + 8 + 4 + 8 = 32 cm.

4) Use the scale factor to get the sides of EFGH (multiply small trapezoid lengths by 3/2):
EF = (3/2) · AB = (3/2) · 12 = 18 cm,
FG = (3/2) · BC = (3/2) · 8 = 12 cm,
GH is given = 6 cm,
HE = (3/2) · AD = (3/2) · 8 = 12 cm.

Perimeter(EFGH) = 18 + 12 + 6 + 12 = 48 cm.

Summary sentences:
- The corresponding side ratios are AB:EF = BC:FG = CD:GH = DA:HE = 2:3.
- The corresponding angles are ∠A ≅ ∠E, ∠B ≅ ∠F, ∠C ≅ ∠G, and ∠D ≅ ∠H.
- The perimeter of trapezoid ABCD is 32 cm, and the perimeter of trapezoid EFGH is 48 cm.