QuestionFind $x$ given that $BC = 2x - 17$ , $AC = 19$ , and $AB = 2x - 12$ .

Studdy Solution

STEP 1

Assumptions1. BC =x -17. AC =193. AB =x -124. The triangle ABC is a right triangle with BC as the hypotenuse.

STEP 2

We can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AC and AB).
$BC^2 = AC^2 + AB^2$

STEP 3

Substitute the given values into the Pythagorean theorem.
$(2x -17)^2 =19^2 + (2x -12)^2$

STEP 4

Expand the squares on both sides of the equation.
$4x^2 -68x +289 =361 +4x^2 -48x +144$

STEP 5

implify the equation by subtracting $4x^2$ from both sides.
$-68x +289 =361 -48x +144$

STEP 6

Combine like terms on both sides of the equation.
$-68x +289 =505 -48x$

STEP 7

Add $48x$ to both sides of the equation to isolate the x terms on one side.
$-20x +289 =505$

STEP 8

Subtract $289$ from both sides of the equation to isolate the x term.
$-20x =216$

STEP 9

Divide both sides of the equation by $-20$ to solve for $x$ .
$x = \frac{216}{-20}$

STEP 10

Calculate the value of $x$ .
$x = -10.8$ However, in the context of this problem, a negative length does not make sense. Therefore, there seems to be a mistake in the problem or our assumptions.

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QuestionFind $x$ given that $BC = 2x - 17$ , $AC = 19$ , and $AB = 2x - 12$ .

Studdy Solution

STEP 1

Assumptions1. BC =x -17. AC =193. AB =x -124. The triangle ABC is a right triangle with BC as the hypotenuse.

STEP 2

We can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AC and AB).
$BC^2 = AC^2 + AB^2$

STEP 3

Substitute the given values into the Pythagorean theorem.
$(2x -17)^2 =19^2 + (2x -12)^2$

STEP 4

Expand the squares on both sides of the equation.
$4x^2 -68x +289 =361 +4x^2 -48x +144$

STEP 5

implify the equation by subtracting $4x^2$ from both sides.
$-68x +289 =361 -48x +144$

STEP 6

Combine like terms on both sides of the equation.
$-68x +289 =505 -48x$

STEP 7

Add $48x$ to both sides of the equation to isolate the x terms on one side.
$-20x +289 =505$

STEP 8

Subtract $289$ from both sides of the equation to isolate the x term.
$-20x =216$

STEP 9

Divide both sides of the equation by $-20$ to solve for $x$ .
$x = \frac{216}{-20}$

STEP 10

Calculate the value of $x$ .
$x = -10.8$ However, in the context of this problem, a negative length does not make sense. Therefore, there seems to be a mistake in the problem or our assumptions.

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QuestionFind xxx given that BC=2x−17BC = 2x - 17BC=2x−17, AC=19AC = 19AC=19, and AB=2x−12AB = 2x - 12AB=2x−12.

Studdy Solution

STEP 1

Assumptions1. BC =x -17. AC =193. AB =x -124. The triangle ABC is a right triangle with BC as the hypotenuse.

STEP 2

We can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AC and AB).BC2=AC2+AB2BC^2 = AC^2 + AB^2BC2=AC2+AB2

STEP 3

Substitute the given values into the Pythagorean theorem.(2x−17)2=192+(2x−12)2(2x -17)^2 =19^2 + (2x -12)^2(2x−17)2=192+(2x−12)2

STEP 4

Expand the squares on both sides of the equation.4x2−68x+289=361+4x2−48x+1444x^2 -68x +289 =361 +4x^2 -48x +1444x2−68x+289=361+4x2−48x+144

STEP 5

implify the equation by subtracting 4x24x^24x2 from both sides.−68x+289=361−48x+144-68x +289 =361 -48x +144−68x+289=361−48x+144

STEP 6

Combine like terms on both sides of the equation.−68x+289=505−48x-68x +289 =505 -48x−68x+289=505−48x

STEP 7

Add 48x48x48x to both sides of the equation to isolate the x terms on one side.−20x+289=505-20x +289 =505−20x+289=505

STEP 8

Subtract 289289289 from both sides of the equation to isolate the x term.−20x=216-20x =216−20x=216

STEP 9

Divide both sides of the equation by −20-20−20 to solve for xxx.x=216−20x = \frac{216}{-20}x=−20216​

STEP 10

Calculate the value of xxx.x=−10.8x = -10.8x=−10.8However, in the context of this problem, a negative length does not make sense. Therefore, there seems to be a mistake in the problem or our assumptions.

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QuestionFind $x$ given that $BC = 2x - 17$ , $AC = 19$ , and $AB = 2x - 12$ .

We can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (BC) is equal to the sum of the squares of the lengths of the other two sides (AC and AB).
$BC^2 = AC^2 + AB^2$

Substitute the given values into the Pythagorean theorem.
$(2x -17)^2 =19^2 + (2x -12)^2$

Expand the squares on both sides of the equation.
$4x^2 -68x +289 =361 +4x^2 -48x +144$

implify the equation by subtracting $4x^2$ from both sides.
$-68x +289 =361 -48x +144$

Combine like terms on both sides of the equation.
$-68x +289 =505 -48x$

Add $48x$ to both sides of the equation to isolate the x terms on one side.
$-20x +289 =505$

Subtract $289$ from both sides of the equation to isolate the x term.
$-20x =216$

Divide both sides of the equation by $-20$ to solve for $x$ .
$x = \frac{216}{-20}$

Calculate the value of $x$ .
$x = -10.8$ However, in the context of this problem, a negative length does not make sense. Therefore, there seems to be a mistake in the problem or our assumptions.