Mathematics Competition For Grade Nine

1) Find the distance between (2,2) and (6,2).أوجد المسافة بين النقطتين

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. In this case, the two points (2,2) and (6,2) have the same y-coordinate, indicating that they lie on the same horizontal line. Since they have different x-coordinates, the distance between them can be calculated as the absolute difference between their x-coordinates, which is 4. Hence, the distance between the two points is 4.

Explanation

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. In this case, the two points (2,2) and (6,2) have the same y-coordinate, indicating that they lie on the same horizontal line. Since they have different x-coordinates, the distance between them can be calculated as the absolute difference between their x-coordinates, which is 4. Hence, the distance between the two points is 4.

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2) Find the distance between (-1,3) and (2,-1).احسب طول المسافة بين النقطتين

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. The formula is d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, the coordinates are (-1, 3) and (2, -1). Plugging these values into the formula, we get d = sqrt((2 - (-1))^2 + (-1 - 3)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. Therefore, the distance between the two points is 5 units.

Explanation

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. The formula is d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, the coordinates are (-1, 3) and (2, -1). Plugging these values into the formula, we get d = sqrt((2 - (-1))^2 + (-1 - 3)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. Therefore, the distance between the two points is 5 units.

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3)

if a = 5, b = 12, then h =

In this question, the values of a and b are given as 5 and 12 respectively. The variable h is not explicitly defined in the question, but based on the given values, it can be inferred that h represents the hypotenuse of a right-angled triangle. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, using the values of a and b, we can calculate the value of h as the square root of (5^2 + 12^2), which is equal to 13.

Explanation

In this question, the values of a and b are given as 5 and 12 respectively. The variable h is not explicitly defined in the question, but based on the given values, it can be inferred that h represents the hypotenuse of a right-angled triangle. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, using the values of a and b, we can calculate the value of h as the square root of (5^2 + 12^2), which is equal to 13.

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4) What is the y-intercept in the following equation:y = -3x + 2

The y-intercept in the equation y = -3x + 2 is the point where the line crosses the y-axis. In this equation, the y-intercept is the value of y when x is equal to 0. By substituting x = 0 into the equation, we get y = -3(0) + 2, which simplifies to y = 2. Therefore, the y-intercept is 2.

Explanation

The y-intercept in the equation y = -3x + 2 is the point where the line crosses the y-axis. In this equation, the y-intercept is the value of y when x is equal to 0. By substituting x = 0 into the equation, we get y = -3(0) + 2, which simplifies to y = 2. Therefore, the y-intercept is 2.

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5) What is the y-intercept in the following equation:y = -5x - 7

The y-intercept is the point where the graph of the equation intersects the y-axis. In the given equation y = -5x - 7, the constant term -7 represents the y-intercept. This means that when x = 0, y will be -7. Therefore, the y-intercept is (-7, -7).

Explanation

The y-intercept is the point where the graph of the equation intersects the y-axis. In the given equation y = -5x - 7, the constant term -7 represents the y-intercept. This means that when x = 0, y will be -7. Therefore, the y-intercept is (-7, -7).

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6) What is the y-intercept in the following equation:y = -47x + 47

The y-intercept in the given equation is 47. This means that the line crosses the y-axis at the point (0, 47). The equation y = -47x + 47 represents a line with a slope of -47 and a y-intercept of 47. The positive sign indicates that the line crosses the y-axis above the origin.

Explanation

The y-intercept in the given equation is 47. This means that the line crosses the y-axis at the point (0, 47). The equation y = -47x + 47 represents a line with a slope of -47 and a y-intercept of 47. The positive sign indicates that the line crosses the y-axis above the origin.

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7) Identify the slope of the line.
y = 3x - 5

The slope of the line can be determined by looking at the coefficient of the x term in the equation. In this case, the coefficient of x is 3, which means that the slope of the line is 3.

Explanation

The slope of the line can be determined by looking at the coefficient of the x term in the equation. In this case, the coefficient of x is 3, which means that the slope of the line is 3.

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8) Identify the y-intercept of the equation.
y = 2/9 x?

The y-intercept of an equation is the value of y when x is equal to 0. In this equation, when x is 0, the value of y is also 0. Therefore, the y-intercept of the equation y = (2/9)x is 0.

Explanation

The y-intercept of an equation is the value of y when x is equal to 0. In this equation, when x is 0, the value of y is also 0. Therefore, the y-intercept of the equation y = (2/9)x is 0.

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9) Find the distance between (-6,-3) and (6,6).احسب طول المسافة بين النقطتين

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. In this case, the distance between (-6,-3) and (6,6) can be calculated as follows:

d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(6 - (-6))^2 + (6 - (-3))^2]
= √[12^2 + 9^2]
= √[144 + 81]
= √225
= 15

Therefore, the distance between the two points is 15 units.

Explanation

The distance between two points in a coordinate plane can be found using the distance formula, which is derived from the Pythagorean theorem. In this case, the distance between (-6,-3) and (6,6) can be calculated as follows:

d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(6 - (-6))^2 + (6 - (-3))^2]
= √[12^2 + 9^2]
= √[144 + 81]
= √225
= 15

Therefore, the distance between the two points is 15 units.

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10)

If a = 3 and b = 4, then h =

The value of h can be determined using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides (a and b). In this case, a = 3 and b = 4. Therefore, h^2 = 3^2 + 4^2 = 9 + 16 = 25. Taking the square root of both sides, we get h = 5.

Explanation

The value of h can be determined using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides (a and b). In this case, a = 3 and b = 4. Therefore, h^2 = 3^2 + 4^2 = 9 + 16 = 25. Taking the square root of both sides, we get h = 5.

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11) Is this the Pythagoras Theorem: a² + b² = c^{2هل هذة نظرية فيثاغورث}

True

False

The given equation, a2 + b2 = c2, is indeed the Pythagorean Theorem. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore, the answer "True" is correct.

Explanation

The given equation, a2 + b2 = c2, is indeed the Pythagorean Theorem. This theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore, the answer "True" is correct.

12) What is the midpoint of A(10, 3) and B(-2, 5)? اكتب إحداثي منتصف المسافة بين النقطتين (A(10, 3) , B(-2, 5

(0, 4)

(4, 4)

(-2, 7)

(0, 3)

(2, 4)

The midpoint of two points can be found by taking the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of point A is 10 and the x-coordinate of point B is -2. Taking the average of these gives us (10 + (-2))/2 = 4. Similarly, the y-coordinate of point A is 3 and the y-coordinate of point B is 5. Taking the average of these gives us (3 + 5)/2 = 4. Therefore, the midpoint of A(10, 3) and B(-2, 5) is (4, 4).

Explanation

The midpoint of two points can be found by taking the average of their x-coordinates and the average of their y-coordinates. In this case, the x-coordinate of point A is 10 and the x-coordinate of point B is -2. Taking the average of these gives us (10 + (-2))/2 = 4. Similarly, the y-coordinate of point A is 3 and the y-coordinate of point B is 5. Taking the average of these gives us (3 + 5)/2 = 4. Therefore, the midpoint of A(10, 3) and B(-2, 5) is (4, 4).

13) Which type of slope is shown?

Negative

Zero Slope

Undefined

The question is asking about the type of slope shown, but it does not provide any information or context to determine the slope. Without any given values or a graph, it is impossible to determine the type of slope. Therefore, the answer is undefined.

Explanation

The question is asking about the type of slope shown, but it does not provide any information or context to determine the slope. Without any given values or a graph, it is impossible to determine the type of slope. Therefore, the answer is undefined.

14) What is the midpoint of C(-5, 3) and D(6, -1)?اكتب إحداثي منتصف المسافة بين النقطتين

(1, 2)

(0.5, 1)

(-0.5, -1)

(5.5, 2)

(11, -5)

The midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the two endpoints. In this case, the x-coordinate of the midpoint is (-5 + 6)/2 = 0.5 and the y-coordinate of the midpoint is (3 + (-1))/2 = 1. Therefore, the midpoint of C(-5, 3) and D(6, -1) is (0.5, 1).

Explanation

The midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the two endpoints. In this case, the x-coordinate of the midpoint is (-5 + 6)/2 = 0.5 and the y-coordinate of the midpoint is (3 + (-1))/2 = 1. Therefore, the midpoint of C(-5, 3) and D(6, -1) is (0.5, 1).

15) Which type of slope is shown?

Positive

Negative

Zero Slope

Undefined

The given correct answer is "Negative." This suggests that the slope shown is decreasing from left to right. In other words, as the x-values increase, the corresponding y-values decrease. This can be visualized as a line that slants downwards from left to right on a graph.

Explanation

The given correct answer is "Negative." This suggests that the slope shown is decreasing from left to right. In other words, as the x-values increase, the corresponding y-values decrease. This can be visualized as a line that slants downwards from left to right on a graph.

16) Which type of slope is shown?

Positive

Negative

Zero Slope

Undefined

The correct answer is positive because the slope of a line is positive when it rises from left to right. In this case, the line is rising as we move from left to right, indicating a positive slope.

Explanation

The correct answer is positive because the slope of a line is positive when it rises from left to right. In this case, the line is rising as we move from left to right, indicating a positive slope.

17) Find the slope of the line though the pair of points.
F(-5, 1), G(0, -9)

2

-2

1/2

-1/2

To find the slope of a line passing through two given points, we can use the formula: slope = (y2 - y1) / (x2 - x1). In this case, the coordinates of point F are (-5, 1) and the coordinates of point G are (0, -9). Plugging these values into the formula, we get: slope = (-9 - 1) / (0 - (-5)) = -10 / 5 = -2. Therefore, the slope of the line passing through points F and G is -2.

Explanation

To find the slope of a line passing through two given points, we can use the formula: slope = (y2 - y1) / (x2 - x1). In this case, the coordinates of point F are (-5, 1) and the coordinates of point G are (0, -9). Plugging these values into the formula, we get: slope = (-9 - 1) / (0 - (-5)) = -10 / 5 = -2. Therefore, the slope of the line passing through points F and G is -2.

18) 4Y-6X=10
What is the slope of this line?

-3/2

6/4

-6/4

3/2

-2/3

The given equation is in the form of y = mx + b, where m represents the slope of the line. By rearranging the equation to the form y = mx + b, we get y = 6/4x + 10/4. Comparing this with the standard form, we can see that the slope is 6/4.

Explanation

The given equation is in the form of y = mx + b, where m represents the slope of the line. By rearranging the equation to the form y = mx + b, we get y = 6/4x + 10/4. Comparing this with the standard form, we can see that the slope is 6/4.

19) Which type of slope is shown?

Positive

Negative

Zero Slope

No Slope

No Slope is the correct answer because the options Positive, Negative, and Zero Slope all indicate some degree of inclination or change in the slope. However, "No Slope" means that the line is completely horizontal and does not have any inclination or change in slope. Therefore, the correct answer is No Slope.

Explanation

No Slope is the correct answer because the options Positive, Negative, and Zero Slope all indicate some degree of inclination or change in the slope. However, "No Slope" means that the line is completely horizontal and does not have any inclination or change in slope. Therefore, the correct answer is No Slope.

20)

According to the Theorem of Pythagoras,

H = (a² + b²)

H² = √(a² + b²)

H = √(a² + b²)

The answer h = √(a² + b²) is correct because it represents the formula derived from the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides (a and b). Therefore, taking the square root of the sum of the squares of a and b gives us the length of the hypotenuse.

Explanation

The answer h = √(a² + b²) is correct because it represents the formula derived from the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (h) is equal to the sum of the squares of the other two sides (a and b). Therefore, taking the square root of the sum of the squares of a and b gives us the length of the hypotenuse.

21)

According to the Theorem of Pythagoras,

A² = h² - b²

A² = h² + b²

A = h - b

The correct answer is a² = h² - b². This is the formula for the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (a) is equal to the difference between the squares of the lengths of the other two sides (h and b).

Explanation

The correct answer is a² = h² - b². This is the formula for the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (a) is equal to the difference between the squares of the lengths of the other two sides (h and b).