Question

Using Pythagoras' Theorem please solve it.​

Answer 1

`\huge\underline\mathcal{Answer \: - }`

Pythagoras theorem states that the square of hypotenuse equals the sum of square of other two sides.

therefore ,

`(25) {}^(2) = (x + 3) {}^(2) + 4 { }^(2) (x + 2) {}^(2) \\ \\ \longrightarrow \: 625 = x {}^(2) + 9 + 6x + 16(x {}^(2) + 4 + 4x) \\ \\ \longrightarrow \: 625 = x {}^(2) + 9 + 6x + 16x {}^(2) + 64x + 64 \\ \\ \longrightarrow \: 625 = 17x {}^(2) + 70x + 73 \\ \\ \longrightarrow \: 17x {}^(2) + 70x + 73 - 625 = 0 \\ \\ \longrightarrow \: 17x {}^(2) + 70x - 552 = 0`

now using quadratic formula ,

`x = \frac{ - b \: \pm \: \sqrt{b {}^(2) - 4ac} }{2a} \\`

here , a = 17 , b = 70 , c = -552

substituting the values in the equation above ,

`x = \frac{70 \: \pm \: \sqrt{70 {}^(2) - 4(17)( - 552)} }{2 * 17} \\`

after simplifying we get ,

`\boxed{x = 4} \: \: or \:\boxed{ x = ( - 138)/(7) } \\`

since length can't be negative , therefore the other value will be neglected and hence ,

`\boxed{value \: of \: x \: = \: 4 \: cm}`

hope helpful ;-;

Answer 2

Answer: x = 4

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Work Shown:

As you mentioned, we'll use Pythagoras' Theorem aka Pythagorean Theorem.

`a^2+b^2 = c^2\\\\(x+3)^2+(4(x+2))^2 = 25^2\\\\(x+3)^2+16(x+2)^2 = 625\\\\(x^2+6x+9)+16(x^2+4x+4) = 625\\\\x^2+6x+9+16x^2+64x+64 = 625\\\\17x^2+70x+73 = 625\\\\17x^2+70x+73-625 = 0\\\\17x^2+70x-552 = 0\\\\`

Next, we'll use the quadratic formula with a = 17, b = 70, c = -552.

`x = (-b\pm√(b^2-4ac))/(2a)\\\\x = (-70\pm√((70)^2-4(17)(-552)))/(2(17))\\\\x = (-70\pm√(42436))/(34)\\\\x = (-70\pm206)/(34)\\\\x = (-70+206)/(34) \ \text{ or } \ x = (-70-206)/(34)\\\\x = (136)/(34) \ \text{ or } \ x = (-276)/(34)\\\\x = 4 \ \text{ or } \ x \approx -8.1176\\\\`

Ignore the negative x value solution. The sides (x+3) and 4(x+2) will be negative values if we plugged in x = -8.1176, but negative side lengths do not make sense.

This makes x = 4 the only possible solution.

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If x = 4, then,

• vertical leg = x+3 = 4+3 = 7
• horizontal leg = 4(x+2) = 4*(4+2) = 4*6 = 24

This is a 7-24-25 right triangle.

We can confirm this using the pythagorean theorem

`a^2+b^2 = c^2\\\\7^2+24^2 = 25^2\\\\49+576 = 625\\\\625 = 625 \ \ \ \checkmark`