the longer leg of a right triangle is 7 cm. more than the shorter leg. The length of the hypotenuse is 3 cm more than twice the length of the shorter lenght

Question

the longer leg of a right triangle is 7 cm. more than the shorter leg. The length of the hypotenuse is 3 cm more than twice the length of the shorter lenght

Answer 1

Assuming that you are solving for the length of each side:

 

Solution:

• Let the shorter leg of the RIGHT triangle equal (x) which is an unknown.
• The longer leg of the RIGHT Triangle is then equal to (x + 7).
• The length of the hypotenuse is then (2x + 3)

Use the Pythagorean Theorem to solve the problem. Assign the values of (a), (b) and (c) to the shorter side, the longer side and the hypotenuse, respectively.

  

1. a^2 + b^2 = c^2  Equation 1.
2. (x)^2 + (x + 7)^2= (2x + 3)^2

• (x)^2 = x^2
• (x + 7)^2 = x^2 + 14x +49
• (2x +3)^2 = 4x^2 + 12x +9

When you insert the above values into Equation 1 and combine like terms you get Equation 2:

 

• -2x^2 + 2x +40 = 0 - Equation 2.  From the standard form of a quadratic equation a = -2, b = 2, c = 40.
• Because the coefficient of the x^2 term is not 1, we will first use decomposition to solve this problem.
• Decompose the 2x term.
• You need to determine two unknown numbers that add to equal (b) and multiply to equal (a)*(c). So select m and n as your unknowns.

                   m + n = b = 2

                   m(n) = a*c = -2 * 40 = -80

                   Those two numbers are 10, -8. Substitute these values into Equation 2.

 

• -2x^2 + (10 - 8)x + 40 = 0 Distribute the (x) term.
• -2x^2 + 10x - 8x + 40 = 0  Group common terms.
• (-2x^2 + 10x) + (-8x + 40) = 0  Factor the GCF
• -2x (x - 5) - 8 (x- 5) = 0  Factor again
• (x - 5) (-2x - 8) = 0 Solve for x.
• x - 5 = 0  or        -2x -8 = 0
• x = 5       or         -2x = 8 implies that x = 8/-2 = -4 (Not a solution because lengths cannot be negative.)

Plug x = 5 back into the equations of the legs of the triangle to find the answers for the lengths:

• Shorter leg a = x = 5.
• Longer leg b = (x + 7) = (5 + 7) = 12.
• Hypotenuse  c = (2x + 3) = (2(5) + 3) = 13.

 

Check the answer by plugging back into the Pythagorean Theorem Equation 1:

• (5)^2 + (12)^2 = (13)^2
• 25 + 144 = 169
• 169 = 169  CHECKS!!!!!!!

       
                                                                                                                     

 

     

Answer 2

A right triangle indicates that we can apply Pythagorean theorem.

 

a² + b² = c²

 

 

Since all the sides of are being compared to the shorter leg,

 

Let shorter leg = x

Let longer leg = x + 7

Let hypotenuse = 2x + 3

 

Now we just use these variables to establish an equation that represents the Pythagorean theorem.

 

x² + (x + 7)² = (2x + 3)²

 

 

Solve for x from this equation.

 

Expand the equation so that it becomes simpler to solve.

 

x² + (x² + 14x + 49) = 4x² + 12x + 9

 

2x² + 14x + 49 = 4x² + 12x + 9

 

 

Move all the terms to the right side so that we have a positive coefficient of x².  Doing this will gives us a quadratic equation that is easier to factor using FOIL.

 

0 = 2x² - 2x - 40

 

 

Factor out a 2 on the right side of the equation.

 

0 = 2(x² - x - 20)

 

x² - x - 20 = 0

 

 

Factor using FOIL.

 

(x - 5)(x + 4) = 0

 

 

Set each factor equal to zero and solve for x from each factor equation.  Accept positive value of x for we cannot have negative dimensions.  Then you will plug in that value of x into the variables to get your actual lengths.