Question

The area of the right triangle shown is 24 square feet. Which equations can be used to find the lengths of the legs of the triangle? Check all that apply.
a
x2 + 2x – 24 = 0

b
x(x + 2) = 24
c
x2 + (x + 2)2 = 100

d
0.5(x)(x + 2) = 24
e
x2 + 2x – 48 = 0

f
x2 + (x + 2)2 = 24

Answer 1

Answer with explanation:

It is given that area of right triangle is 24 square feet

Since Sides of any triangle is always Positive.So, we have to find those equation among six options , which gives two positive Integral values,as area of right triangle is equal to product of base length and Corresponding Altitude.

Starting from option 1

⇒1. x² +2 x - 24 =0

 Splitting the middle term

→x²+6 x - 4 x -24=0

→x×(x+6)-4×(x+6)=0  

→(x-4)(x+6)

By substituting the expression equal to , 0, we will get one positive zero and one negative Zero so , it can't be the equation which can be sides of right triangle

Option B:

→x×(x+2)=0

Gives, two zeroes which are equal to 0 and , -2 So, it is also not that expression which gives sides of right triangle, when area =24 square feet

Option C:

x²+ (x+2)×2=100

x²+2 x +4-100=0

x²+2 x-96=0

$x=\frac{-2 +\sqrt{2^2\pm 4 \times 96}}{2}\\\\x=\frac{-2 \pm\sqrt{4+384}}{2}$

It will give one Positive zero and one negative zero ,so it can't be the equation which gives length of legs of right triangle.

Option D:

0.5 × x×(x+2)=24,

as 0.5 is equal to $\frac{1}{2}$

x²+2 x -48=0

(x-6)(x+8)=0

The two zeroes are , such that one of them is positive and other is negative.So this is not the equation which can be used to find the lengths of the legs of the right triangle.

⇒Option E.

x²+2 x - 48=0

Same as option D

It cant be the expression which can be used to find the lengths of the legs of the triangle.

⇒Option F:

x²+(x+2)×2=24

x²+2 x +4-24=0

x²+ 2 x -20=0

$x=\frac{-2\pm \sqrt{2^2-4 \times (-20)\times 1}}{2}$

It will give one positive zero and one negative zero , so it can't be the equation which can be used to find the lengths of the legs of the triangle.

→→None, of the option can be used to find the length of sides of right  triangle when area is equal to 24 square feet because none of them gives two positive Integral value.

Answer 2

Not sure, but the areas of a triangle is BH/2 so the dimensions should be 8 and 6 then 10.
8 x 6 = 48
48/2 = 24
Maybe that can help with the equation