Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 months ago

# The base of a right triangle is four more than three times the height. If the hypotenuse is five times the...?

height find how high the triangle is (the height)

Relevance
• Let a=the base; b=the height; c=the hypotenuse. It follows that

a^2+b^2=c^2

=>

(3b+4)^2+b^2=(5b)^2

=>

9b^2+24b+16+b^2=25b^2

=>

15b^2-24b-16=0

=>

b=2.106 or b=-0.506 (rejected)

=>

the height=2.106 approximately.

• The more obvious case has already been covered, but there is another solution, equally valid and quite a bit simpler.

Do you understand that any side of a triangle may be identified as the base? Let the hypotenuse be the base. The height then is the altitude from the angle opposite the hypotenuse. The fact that it is a right triangle is of no matter, but does no harm.

Let the height be h.

(hypotenuse) = 4 + 3h

(hypotenuse) = 5h

5h = 4 + 3h

2h = 4

h = 2

The height of the triangle is 2.

• Refer to the following figure :

According to the question --

( 3 h + 4 )² + h²  = ( 5 h )² ................. Using Pythagoras theorem.

( 3 h + 4 )² = ( 5 h )² -  h²

=> (3h+4)²  =  24 h²   ........ Take squire root of both the sides.

=> 3 h + 4  =  ± 2√6 h  ..................

We will, now, calculate h  considering firstly +ve and again considering (-)ve sign.

(1) Positive sign :  3 h + 4  =  + 2√6 h

=>  h (2√6 - 3)  =  4  ====>>>>  h  =  4/(2√6 - 3) =  2.27

(2) Negative sign. :  3 h + 4  =  -  2√6 h

=>  3 h +  2√6 h  =  - 4  =====>>>>  h = -ve value.... to be neglected.

Hence  h  =  2.27  ........... Answer • The base of a right triangle is 4 more than 3 times the height.

If the hypotenuse is 5 times the height find how high the triangle is (the height)

h^2 + (3h + 4)^2 = (5h)^2

10h^2 + 24h + 16 = 25h^2

15h^2 - 24h - 16 = 0

75/128 (h - 4/5)^2 = 1

Solutions:

h = 4/5 - (8 sqrt(2/3))/5 ≈ -0.50639

h = 4/5 + (8 sqrt(2/3))/5 ≈ 2.1064      ====> Answer