Line segments bar(QR) and bar(ST) are both diameters of a circle. This circle passes through the point (3,13.2). If bar(QR) lies on the line whose equation is y=2.5 x+4.9 and if bar(ST) lies on the line whose equation is y=3x+2.4, what is the area of the circle?

Find Intersection Point: First, we need to find the intersection point of the two lines y=2.5x+4.9 and y=3x+2.4, which will give us the center of the circle. To do this, we set the two equations equal to each other and solve for x: 2.5x + 4.9 = 3x + 2.4Calculate x-coordinate: Subtract 2.5x from both sides to get: 4.9 = 0.5x + 2.4Calculate y-coordinate: Subtract 2.4 from both sides to get: 2.5 = 0.5xFind Radius: Divide both sides by 0.5 to find x: x = 2.5 / 0.5 x = 5Calculate Distance: Now that we have the x-coordinate of the center, we can substitute it back into one of the original equations to find the y-coordinate of the center. Let's use the first equation: y = 2.5(5) + 4.9Calculate Radius: Calculate the y-coordinate: y = 12.5 + 4.9 y = 17.4Calculate Area: The center of the circle is at the point (5, 17.4). Now we need to find the radius of the circle by calculating the distance between the center and the given point on the circle (3, 13.2) using the distance formula: radius = √((x2 - x1)² + (y2 - y1)²)Substitute Radius: Substitute the coordinates into the distance formula: radius = √((3 - 5)² + (13.2 - 17.4)²)Calculate Area: Calculate the radius: radius = √((-2)² + (-4.2)²) radius = √(4 + 17.64) radius = √21.64Round Area: Find the numerical value of the radius: radius ≈ 4.65Now that we have the radius, we can calculate the area of the circle using the formula: Area = π * radius²Substitute the radius into the area formula: Area ≈ π * (4.65)²Calculate the area: Area ≈ π * 21.6225 Area ≈ 67.94775πRound the area to a reasonable number of decimal places, if necessary. However, since the problem does not specify, we can leave the answer in terms of π: Area ≈ 67.95π square units

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Line segments bar(QR) and bar(ST) are both diameters of a circle. This circle passes through the point (3,13.2). If bar(QR) lies on the line whose equation is y=2.5 x+4.9 and if
bar(ST) lies on the line whose equation is y=3x+2.4, what is the area of the circle?

Line segments $\overline{Q R}$ and $\overline{S T}$ are both diameters of a circle. This circle passes through the point $(3,13.2)$ . If $\overline{Q R}$ lies on the line whose equation is $y=2.5 x+4.9$ and if $\overline{S T}$ lies on the line whose equation is $y=3 x+2.4$ , what is the area of the circle?

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Grade 6

Area of quadrilaterals and triangles: word problems

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Q. Line segments

\overline{Q R}

and

\overline{S T}

are both diameters of a circle. This circle passes through the point

(3,13.2)

. If

\overline{Q R}

lies on the line whose equation is

y=2.5 x+4.9

and if

\overline{S T}

lies on the line whose equation is

y=3 x+2.4

, what is the area of the circle?

Find Intersection Point: First, we need to find the intersection point of the two lines $y=2.5x+4.9$ and $y=3x+2.4$ , which will give us the center of the circle. $\newline$ To do this, we set the two equations equal to each other and solve for $x$ : $\newline$ $2.5x + 4.9 = 3x + 2.4$
Calculate x-coordinate: Subtract $2.5x$ from both sides to get: $\newline$ $4.9 = 0.5x + 2.4$
Calculate y-coordinate: Subtract $2.4$ from both sides to get: $\newline$ $2.5 = 0.5x$
Find Radius: Divide both sides by $0.5$ to find $x$ : $\newline$ $x = \frac{2.5}{0.5}$ $\newline$ $x = 5$
Calculate Distance: Now that we have the $x$ -coordinate of the center, we can substitute it back into one of the original equations to find the $y$ -coordinate of the center. Let's use the first equation: $\newline$ $y = 2.5(5) + 4.9$
Calculate Radius: Calculate the y-coordinate: $\newline$ y = $12.5 + 4.9$ $\newline$ y = $17.4$
Calculate Area: The center of the circle is at the point $(5, 17.4)$ . Now we need to find the radius of the circle by calculating the distance between the center and the given point on the circle $(3, 13.2)$ using the distance formula: $\newline$ $\text{radius} = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$
Substitute Radius: Substitute the coordinates into the distance formula: $\newline$ radius = $\sqrt{(3 - 5)^2 + (13.2 - 17.4)^2}$
Calculate Area: Calculate the radius: $\newline$ radius = $\sqrt{(-2)^2 + (-4.2)^2}$ $\newline$ radius = $\sqrt{4 + 17.64}$ $\newline$ radius = $\sqrt{21.64}$
Round Area: Find the numerical value of the radius: $\text{radius} \approx 4.65$
Round Area: Find the numerical value of the radius: $\newline$ radius $\approx 4.65$ Now that we have the radius, we can calculate the area of the circle using the formula: $\newline$ Area $= \pi \cdot \text{radius}^2$
Round Area: Find the numerical value of the radius: $\newline$ radius $\approx 4.65$ Now that we have the radius, we can calculate the area of the circle using the formula: $\newline$ Area $= \pi \times \text{radius}^2$ Substitute the radius into the area formula: $\newline$ Area $\approx \pi \times (4.65)^2$
Round Area: Find the numerical value of the radius: $\newline$ radius $\approx 4.65$ Now that we have the radius, we can calculate the area of the circle using the formula: $\newline$ Area $= \pi \times \text{radius}^2$ Substitute the radius into the area formula: $\newline$ Area $\approx \pi \times (4.65)^2$ Calculate the area: $\newline$ Area $\approx \pi \times 21.6225$ $\newline$ Area $\approx 67.94775\pi$
Round Area: Find the numerical value of the radius: $\newline$ $radius \approx 4.65$ Now that we have the radius, we can calculate the area of the circle using the formula: $\newline$ $Area = \pi \times radius^2$ Substitute the radius into the area formula: $\newline$ $Area \approx \pi \times (4.65)^2$ Calculate the area: $\newline$ $Area \approx \pi \times 21.6225$ $\newline$ $Area \approx 67.94775\pi$ Round the area to a reasonable number of decimal places, if necessary. However, since the problem does not specify, we can leave the answer in terms of $\pi$ : $\newline$ $Area \approx 67.95\pi$ square units