SOLUTION: The diagonal of a rectangle has a length 50. If its legs are in the ratio 3:4 , what is its perimeter? Thank you!

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Question 922349: The diagonal of a rectangle has a length 50. If its legs are in the ratio 3:4 , what is its perimeter? Thank you!
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

The diagonal d of a rectangle has a length 50: then
d=50

If its legs a and b are in the ratio 3%3A4 , then we have:
a%3Ab=3%3A4 ...solve for a
4a=3b
a=3b%2F4
a=%283%2F4%29b


what is its perimeter?
the diagonal of a rectangle along with the legs a and b forms a right angle triangle; so, usu Pythagorean theorem to find the length of the legs
d%5E2=a%5E2%2Bb%5E2 ....substitute a=%283%2F4%29b and d=50
50%5E2=%28%283%2F4%29b%29%5E2%2Bb%5E2

2500=9b%5E2%2F16%2Bb%5E2 ...solve for b

2500=9b%5E2%2F16%2B16b%5E2%2F16

2500=25b%5E2%2F16

2500%2A16=25b%5E2

%282500%2A16%29%2F25=b%5E2

%28cross%282500%29100%2A16%29%2Fcross%2825%29=b%5E2

100%2A16=b%5E2

1600=b%5E2

b=sqrt%281600%29

highlight%28b=40%29 or highlight%28b=-40%29
since we have a length, we don't need negative solution;
so, highlight%28b=40%29
now go back to d%5E2=a%5E2%2Bb%5E2, plug in d=50 and b=40 and calculate a
50%5E2=a%5E2%2B40%5E2
2500=a%5E2%2B1600
2500-1600=a%5E2
900=a%5E2
sqrt%28900%29=a
highlight%28a=30%29
now we know the length of the sides and we can calculate perimeter P:
P=2a%2B2b
P=2%2A30%2B2%2A40
P=60%2B80
highlight%28P=140%29

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

The diagonal of a rectangle has a length 50. If its legs are in the ratio 3:4 , what is its perimeter? Thank you!


Let the multiplicative factor be x

Then the shorter leg measures 3x, and the longer leg measures 4x

Using the pythagorean formula, we get: 

%283x%29%5E2+%2B+%284x%29%5E2+=+50%5E2

9x%5E2+%2B+16x%5E2+=+2500

25x%5E2+=+2500

25%28x%5E2%29+=+25%28100%29

x%5E2+=+100

x, or multiplicative factor = sqrt%28100%29, or 10



Measure of shorter leg: 3(10), or 30 units

Measure of longer leg:  4(10), or 40 units

Perimeter of rectangle: 2(30 + 40), or 2(70), or highlight_green%28140%29 units