If I Have A Rectangle And The Diagonal Is 27" And The Height/width Ratio Is 4:3, Then What Are The Height And Width?

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2 Answers

Anonymous Profile
Anonymous answered
If Width = 4/3 Height, then you can substitute into the Pythagorean theorem and solve.  

Pythagoras tells us the length of the Diagonal2 is equal to the Height2 x Width2

272 = Height2 x (4/3 Height)2

729 = Height2 x (4/3)2 (Height)2

729 = (1 + 16/9) (Height)2

729/(1+ 16/9) = Height2

Height = √(729/(1+16/9))

Height = 16.2

Width = 4/3 (16.2)

Width = 21.6

Now let's check it;

16.22 + 21.62 = ?

16.22 + 21.62 = 729

√729 = 27

So it checks.  You could also solve this using trig relationships.
thanked the writer.
Oddman
Oddman commented
The 3:4:5 triangle is one of the first that you learn about with respect to the Pythagorean Theorem. If the sides have ratio 4:3, the diagonal is 5 "ratio units." Thus the height and width are 4/5 and 3/5 of the diagonal measure.
Anonymous Profile
Anonymous answered
As Oddman stated, the 3,4,5 is one of the first Pythagorean triangles generally studied.    I am guessing since you were given the height and width as ratios, you may be expected to answer the question as such, so if this is the case .... Here goes.    Given ... Diagonal - 27"  ....[This is the 5 of the 3,4,5 triangle]    ... Height:Width    4:3    Step 1:  Divide Diagonal value (27) by 5 to get [1 unit] ..... 27/5   = 5.4"    Step 2:  Find Height ... Multiply your 1 unit by 4 ..... 5.4 x 4 = 21.6"    Step 3: Find Width .... Multiply you 1 unit by 3 ... 5.4 x 3 = 16.2"    Your diagonal is 27"  .... Given  Your Height is 21.5" .... Calculated from ratio  Your Width is 16.2" .... Calculated from ratio.    I hope this helps.

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