Math / Geometry

Question2. If $\triangle A B C$ is a right triangle formed by the placement of three squares as shown below, which is the closest in square inches (in ${ }^{2}$ ) to the area of the shaded square?

Studdy Solution

STEP 1

1. The triangle $\triangle ABC$ is a right triangle.
2. The squares are placed on each side of the triangle.
3. The side $AB$ has a length of 15.5 inches.
4. The area of the square on side $BC$ is 329.5 square inches.
5. We need to find the area of the shaded square on the hypotenuse $AC$ .

STEP 2

1. Calculate the length of side $BC$ .
2. Use the Pythagorean theorem to find the length of the hypotenuse $AC$ .
3. Calculate the area of the shaded square on the hypotenuse.

STEP 3

To find the length of side $BC$ , use the area of the square on $BC$ .
The area of a square is given by the formula $\text{Area} = \text{side}^2$ .
Let $x$ be the length of side $BC$ .
$x^2 = 329.5$

STEP 4

Solve for $x$ by taking the square root of both sides:
$x = \sqrt{329.5}$

STEP 5

Use the Pythagorean theorem to find the length of the hypotenuse $AC$ .
The Pythagorean theorem states that for a right triangle, $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
Let $c$ be the length of the hypotenuse $AC$ .
$(15.5)^2 + (\sqrt{329.5})^2 = c^2$

STEP 6

Calculate $(15.5)^2$ :
$(15.5)^2 = 240.25$

STEP 7

Substitute the values into the Pythagorean theorem:
$240.25 + 329.5 = c^2$

STEP 8

Add the values:
$569.75 = c^2$

STEP 9

Solve for $c$ by taking the square root of both sides:
$c = \sqrt{569.75}$

STEP 10

Calculate the area of the shaded square on the hypotenuse $AC$ .
The area of the square is given by the formula $\text{Area} = \text{side}^2$ .
$\text{Area of shaded square} = (\sqrt{569.75})^2$

STEP 11

Since $(\sqrt{569.75})^2 = 569.75$ , the area of the shaded square is:
$\text{Area of shaded square} = 569.75$
The area of the shaded square is approximately:
$\boxed{569.75} \text{ square inches}$

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Question2. If $\triangle A B C$ is a right triangle formed by the placement of three squares as shown below, which is the closest in square inches (in ${ }^{2}$ ) to the area of the shaded square?

Studdy Solution

STEP 1

1. The triangle $\triangle ABC$ is a right triangle.
2. The squares are placed on each side of the triangle.
3. The side $AB$ has a length of 15.5 inches.
4. The area of the square on side $BC$ is 329.5 square inches.
5. We need to find the area of the shaded square on the hypotenuse $AC$ .

STEP 2

1. Calculate the length of side $BC$ .
2. Use the Pythagorean theorem to find the length of the hypotenuse $AC$ .
3. Calculate the area of the shaded square on the hypotenuse.

STEP 3

To find the length of side $BC$ , use the area of the square on $BC$ .
The area of a square is given by the formula $\text{Area} = \text{side}^2$ .
Let $x$ be the length of side $BC$ .
$x^2 = 329.5$

STEP 4

Solve for $x$ by taking the square root of both sides:
$x = \sqrt{329.5}$

STEP 5

Use the Pythagorean theorem to find the length of the hypotenuse $AC$ .
The Pythagorean theorem states that for a right triangle, $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
Let $c$ be the length of the hypotenuse $AC$ .
$(15.5)^2 + (\sqrt{329.5})^2 = c^2$

STEP 6

Calculate $(15.5)^2$ :
$(15.5)^2 = 240.25$

STEP 7

Substitute the values into the Pythagorean theorem:
$240.25 + 329.5 = c^2$

STEP 8

Add the values:
$569.75 = c^2$

STEP 9

Solve for $c$ by taking the square root of both sides:
$c = \sqrt{569.75}$

STEP 10

Calculate the area of the shaded square on the hypotenuse $AC$ .
The area of the square is given by the formula $\text{Area} = \text{side}^2$ .
$\text{Area of shaded square} = (\sqrt{569.75})^2$

STEP 11

Since $(\sqrt{569.75})^2 = 569.75$ , the area of the shaded square is:
$\text{Area of shaded square} = 569.75$
The area of the shaded square is approximately:
$\boxed{569.75} \text{ square inches}$

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Question2. If △ABC\triangle A B C△ABC is a right triangle formed by the placement of three squares as shown below, which is the closest in square inches (in 2{ }^{2}2 ) to the area of the shaded square?

Studdy Solution

STEP 1

STEP 2

1. Calculate the length of side BC BC BC.2. Use the Pythagorean theorem to find the length of the hypotenuse AC AC AC.3. Calculate the area of the shaded square on the hypotenuse.

STEP 3

To find the length of side BC BC BC, use the area of the square on BC BC BC.The area of a square is given by the formula Area=side2 \text{Area} = \text{side}^2 Area=side2.Let x x x be the length of side BC BC BC.x2=329.5 x^2 = 329.5 x2=329.5

STEP 4

Solve for x x x by taking the square root of both sides:x=329.5 x = \sqrt{329.5} x=329.5​

STEP 5

STEP 6

Calculate (15.5)2 (15.5)^2 (15.5)2:(15.5)2=240.25 (15.5)^2 = 240.25 (15.5)2=240.25

STEP 7

Substitute the values into the Pythagorean theorem:240.25+329.5=c2 240.25 + 329.5 = c^2 240.25+329.5=c2

STEP 8

Add the values:569.75=c2 569.75 = c^2 569.75=c2

STEP 9

Solve for c c c by taking the square root of both sides:c=569.75 c = \sqrt{569.75} c=569.75​

STEP 10

Calculate the area of the shaded square on the hypotenuse AC AC AC.The area of the square is given by the formula Area=side2 \text{Area} = \text{side}^2 Area=side2.Area of shaded square=(569.75)2 \text{Area of shaded square} = (\sqrt{569.75})^2 Area of shaded square=(569.75​)2

STEP 11

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Question2. If $\triangle A B C$ is a right triangle formed by the placement of three squares as shown below, which is the closest in square inches (in ${ }^{2}$ ) to the area of the shaded square?

1. Calculate the length of side $BC$ .
2. Use the Pythagorean theorem to find the length of the hypotenuse $AC$ .
3. Calculate the area of the shaded square on the hypotenuse.

To find the length of side $BC$ , use the area of the square on $BC$ .
The area of a square is given by the formula $\text{Area} = \text{side}^2$ .
Let $x$ be the length of side $BC$ .
$x^2 = 329.5$

Solve for $x$ by taking the square root of both sides:
$x = \sqrt{329.5}$

Calculate $(15.5)^2$ :
$(15.5)^2 = 240.25$

Substitute the values into the Pythagorean theorem:
$240.25 + 329.5 = c^2$

Add the values:
$569.75 = c^2$

Solve for $c$ by taking the square root of both sides:
$c = \sqrt{569.75}$

Calculate the area of the shaded square on the hypotenuse $AC$ .
The area of the square is given by the formula $\text{Area} = \text{side}^2$ .
$\text{Area of shaded square} = (\sqrt{569.75})^2$