In the figure, ABCD is a rectangle, AG=15, BG=6, and AD=28. Determine the area of rectangle EFGH.

Question

Here is the link to the figure. https://docs.google.com/document/d/1_v716ZbsRboCHtT6HR38yctLmMuBDtGiKfVgbQ3ieyc/edit?usp=sharing

Answer 1

The picture does help.

problem: find the area of rectangle EFGH

Givens:

1. rectangle ABCD
2. AG=15
3. BG=6
4. AD=28
5. all rectangles have 4-right angles (90 degrees)
6. all triangles have a total of 180 degreeswhen the 3 angles are added

Now we start solving:

AD=BC=28

AB=AG+BG=15+6=21

Pythagoras give us C² : 21²+28²=1225, so AC=35

21-28-35 is a 3-4-5 triangle (divide by 7), so it's angles are 90+36.87+53.13 degrees.

Now it gets easier. Looking at the picture you know that:

1. angle AFG is 90 degrees
2. angle GAF is 53.13 degrees
3. THEREFORE, angle FGA MUST be 36.87
4. that makes triangle AFG a 3-4-5 triangle!

AND

1. angle GBH is 90 degrees
2. angle BGH has to be 180-90-36.87=53.13 degrees
3. angle BHG has to be 180-90-53.13=36.87
4. BOOM! triangle BGH is ALSO a 3-4-5 triangle

Back to the givens:

1. AG=15
2. BG=6

In triangle AFG, AG is in the 5 position, and equal to 15 (5x3). SO:

1. GF is the 4 position, so it must be 4x3=12 **keep this**
2. AF is the 3 position, but we don't need it

In triangle BGH, BG is in the 3 position, and equal to 6 (3x2). SO:

1. GH is the 5 position, so it must be 5x2=10 **keep this**
2. BH is the 4 position, but we don't need it

Double BOOM! We now have 2 sides of the rectangle: GF=12 and GH=10.

Area EFGH= 10x12=120 square.