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Question

The shortest leg of a triangle is 7 centimeters shorter than the other leg. The hypotenuse of this triangle is 17 centimeters. What are the lengths of the two legs of this triangle?

 

 

The shortest leg is _____ centimeters long

 

The other leg is _____ centimeters long

Answer 1

the triangle is a right angle triangle

 

let the longest leg = x

 

The shortest leg = x-7cm, since it is 7cm shorter that the longest leg

 

The Hypotenuse is 17 cm

 

Therefore in a right angle triangle

 

17² = x² + ( x-7)² ,  289 = ( x² + x² -14x + 49) = (2x² -14x +49)

 

Rearranging,  2x²-14x +49 -289 = 0

 

2x²-14x -240 = 0  Divide both sides by 2, x²-7x- 120 =0  

 

factoring ( x-15)(x+8)=0, the two roots are x=15, and x=-8

 

Therefore the long side x=15cm, the short side is x-7 = 15-7 = 8cm

 

Proof:  the hypotenuse is 17cm = √(15²+8²) = 17cm

Answer 2

The hypotenuse by definition is the side of a triangle that is opposite of a right angle.

 

Draw a diagram of a triangle. Label the two legs "a" and "b" and label the hypotenuse "c".

 

It doesn't matter which leg ("a" or "b") is the shorter of the two, just pick one.

I will pick "a" as the shorter one.

 

Label a = x - 7, b= x and c = 17

 

The relationship of the side lengths of a right triangle is a^2 + b^2 = c^2. Plug in these values into the equation

 

a^2 + b^2 = c^2

(x - 7)^2 + (x)^2 = (17)^2

(x^2 - 14x + 49) + (x^2) = 289

2x^2 - 14x + 49 = 289

2x^2 - 14x - 240 = 0

x^2 - 7x - 120 = 0

(x - 15) (x + 8) = 0

 

x - 15 = 0

x = 15

 

x + 8 = 0

x = -8

 

 

 

A distance cannot equal a negative number, therefore, x must equal 15

 

The shortest leg is 8 centimeters long, and the other leg is 15 centimeters long.