These are 8 questions and 8 answers.
Page 1:
1) Triangle ABC has vertices A(-2,1), B(0,5), and C (8,1). What type of triangle is triangle ABC
Answer: option A. acute triangle.
Explanation:
Determine the lengths of the three sides:
Side AB: √[(0+2)^2 + (5-1)^2 ] = √ [4 + 16] = √(20)
Side BC: √ [ (8-0)^2 + (1 -5)^2 ] = √ [ 64 + 16 ] = √(86)
Side AC: √ [ (8 + 2)^2 + (1 - 1)^2 ] = √100 = 10
Compare the square of the side length of the largest side with the sum of the squares of the other two sides.:
AC^2 = 100
BC^2 = 86
AB^2 = 20
BC^2 + AB^2 = 86 + 20 = 106
100 < 106 => AC^2 < BC^2 + AB^2 =>
all the angles are less than 90°, which means the triangle is actue.
That is becasue when the square of the largest side is equal to the sum of the squares of the smaller sides, there is a right angle and the triangle is a right triangle (Pytaghorean theorem). When the square of the largest side is less than the sum of the squares of the other two sides, the angle is less than 90°, and the triangle is acute.
2) Which set represents the side lengths of a right trianle?
Answer: option C: 9, 12, 15
Explanation:
Check whic set obeys Pythagorean theorem.
15^2 = 225
12^2 = 144
9^2 = 81
.
144 + 81 = 225 => 15^2 = 12^2 + 9^2 => right triangle
3) Which property disproves the statement in the student's notes?
Answer: option B) all angles of a rhombus are not congruent.
Explanation:
A rhombus is a quadrilateral whose four sides all have the same length. It is a quadrilateral.
The angles do not need to be congruent. Only the square, which is only one special case of rhombus, has the four angles congruent, but the masure of the angles of other rhombus are not equal.
4) What is the are of the figure?
Answer: 85 unit^2
Explanation:
1) Area of the rectangle = base * height = 4 * 9 = 36
2) area of the trapezoid = height * half of the sum of the bases = 7 * (3+7)/2 = 35
3) area of the rhombus = half the product of the diagonals = 7 * 4 / 2 = 14
Result: 36 + 35 + 14 = 85
That option is not among the choices. The nearest one is 84 unit^2.
Page 2 is the same page 1.
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5) Question 8:
What line can be used to draw the altitude of the triangle FGH
Answer: option D: 3y = 7x + 16
Explanation:
1) All triangles have 3 altitudes (one for each vertex).
2) Now I am going to find the altitude of the vertex H (respect to base FG.
3) First calculate the slope of the segment FG:
slope = [y2 - y1] / [x2 - x1]
x2, y2 are the coordinates of the point G (6, -2) and x1, y1 are the coordinates of the point F (-8,4)
=> slope = [ -2 - 4] / [6 - (-8) ] = [-6] / [14] = - 3/7
4) Now calculate the slope of the perpendicular line (because the altitude is perpendicular to the base).
The lines of perpendicular lines obey the rule: slope line A = - slope line B / 2
So, the slope of the line perpendicular to the base is - [1 / (-3/7) ] = 7/3
5) The equation of the line with slope 3/7 that contains the point H (-4, - 4) is:
y - (-4) 7
--------- = -----
x - (-4) 3
=> 3(y + 4) = 7(x + 4)
=> 3y + 12 = 7x + 28
=> 3y = 7x + 28 - 12
=> 3y = 7x + 16 which is the answer
6) Question 8. How will be the volume affected
Answer: option c. the volume will be 9 times larger.
Explanation:
1) Formula of the volume of rectagular prism
Volume = length * width * height
2) Name, V the volume, l the length, w the width, h the heigth => V = l * w * h
3) Triple the length and the width:
New volume = 3l * 3w * h = 9 l * w * h = 9V
4) You have proved that the new volumen is 9 times the original volume, which is the option c.
7) Question 9
What is the approximate length of the median of the trapezoid ABCD.
Answer: option b. 7.8 units
Explanation:
1) The median of a trapezoid is half the sum of the two parallel segments
2) The two parallel segments are AB and CD
3) The length of the segment AB is:
AB^2 = (3+7)^2 + (7-2)^2 = 10^2 + 5^2 = 100 + 25 = 125
=> AB = √125 ≈ 11.18
4) The length of the sement CD is:
CD^2 = (3 + 1)^2 + (1+1)^2 = 4^2 + 2^2 = 16 + 4 = 20
=> CD = √20 ≈ 4.47
5) The median is [11.48 + 4.47] / 2 ≈ 7.83 => option b.
8) Question 10. Problem
Answer: option B. 34.8 feet
Explanation:
1) Variables:
Ha = 4 ft (heigth of Abigail)
α = 38°
s = 50 ft (length of the string)
Hk = ? (altitude of the kite)
2) formula of sine ratio:
sin α = opposite-leg / adjacent leg
3) solution
sin α = x / s => x = sinα * s = sin(38°) * 50 ft = 30.78 ft
Hk = x + Ha = 30.78 ft + 4 ft = 34.78 ft => aption B. 34.8 feet
Page 1:
1) Triangle ABC has vertices A(-2,1), B(0,5), and C (8,1). What type of triangle is triangle ABC
Answer: option A. acute triangle.
Explanation:
Determine the lengths of the three sides:
Side AB: √[(0+2)^2 + (5-1)^2 ] = √ [4 + 16] = √(20)
Side BC: √ [ (8-0)^2 + (1 -5)^2 ] = √ [ 64 + 16 ] = √(86)
Side AC: √ [ (8 + 2)^2 + (1 - 1)^2 ] = √100 = 10
Compare the square of the side length of the largest side with the sum of the squares of the other two sides.:
AC^2 = 100
BC^2 = 86
AB^2 = 20
BC^2 + AB^2 = 86 + 20 = 106
100 < 106 => AC^2 < BC^2 + AB^2 =>
all the angles are less than 90°, which means the triangle is actue.
That is becasue when the square of the largest side is equal to the sum of the squares of the smaller sides, there is a right angle and the triangle is a right triangle (Pytaghorean theorem). When the square of the largest side is less than the sum of the squares of the other two sides, the angle is less than 90°, and the triangle is acute.
2) Which set represents the side lengths of a right trianle?
Answer: option C: 9, 12, 15
Explanation:
Check whic set obeys Pythagorean theorem.
15^2 = 225
12^2 = 144
9^2 = 81
.
144 + 81 = 225 => 15^2 = 12^2 + 9^2 => right triangle
3) Which property disproves the statement in the student's notes?
Answer: option B) all angles of a rhombus are not congruent.
Explanation:
A rhombus is a quadrilateral whose four sides all have the same length. It is a quadrilateral.
The angles do not need to be congruent. Only the square, which is only one special case of rhombus, has the four angles congruent, but the masure of the angles of other rhombus are not equal.
4) What is the are of the figure?
Answer: 85 unit^2
Explanation:
1) Area of the rectangle = base * height = 4 * 9 = 36
2) area of the trapezoid = height * half of the sum of the bases = 7 * (3+7)/2 = 35
3) area of the rhombus = half the product of the diagonals = 7 * 4 / 2 = 14
Result: 36 + 35 + 14 = 85
That option is not among the choices. The nearest one is 84 unit^2.
Page 2 is the same page 1.
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5) Question 8:
What line can be used to draw the altitude of the triangle FGH
Answer: option D: 3y = 7x + 16
Explanation:
1) All triangles have 3 altitudes (one for each vertex).
2) Now I am going to find the altitude of the vertex H (respect to base FG.
3) First calculate the slope of the segment FG:
slope = [y2 - y1] / [x2 - x1]
x2, y2 are the coordinates of the point G (6, -2) and x1, y1 are the coordinates of the point F (-8,4)
=> slope = [ -2 - 4] / [6 - (-8) ] = [-6] / [14] = - 3/7
4) Now calculate the slope of the perpendicular line (because the altitude is perpendicular to the base).
The lines of perpendicular lines obey the rule: slope line A = - slope line B / 2
So, the slope of the line perpendicular to the base is - [1 / (-3/7) ] = 7/3
5) The equation of the line with slope 3/7 that contains the point H (-4, - 4) is:
y - (-4) 7
--------- = -----
x - (-4) 3
=> 3(y + 4) = 7(x + 4)
=> 3y + 12 = 7x + 28
=> 3y = 7x + 28 - 12
=> 3y = 7x + 16 which is the answer
6) Question 8. How will be the volume affected
Answer: option c. the volume will be 9 times larger.
Explanation:
1) Formula of the volume of rectagular prism
Volume = length * width * height
2) Name, V the volume, l the length, w the width, h the heigth => V = l * w * h
3) Triple the length and the width:
New volume = 3l * 3w * h = 9 l * w * h = 9V
4) You have proved that the new volumen is 9 times the original volume, which is the option c.
7) Question 9
What is the approximate length of the median of the trapezoid ABCD.
Answer: option b. 7.8 units
Explanation:
1) The median of a trapezoid is half the sum of the two parallel segments
2) The two parallel segments are AB and CD
3) The length of the segment AB is:
AB^2 = (3+7)^2 + (7-2)^2 = 10^2 + 5^2 = 100 + 25 = 125
=> AB = √125 ≈ 11.18
4) The length of the sement CD is:
CD^2 = (3 + 1)^2 + (1+1)^2 = 4^2 + 2^2 = 16 + 4 = 20
=> CD = √20 ≈ 4.47
5) The median is [11.48 + 4.47] / 2 ≈ 7.83 => option b.
8) Question 10. Problem
Answer: option B. 34.8 feet
Explanation:
1) Variables:
Ha = 4 ft (heigth of Abigail)
α = 38°
s = 50 ft (length of the string)
Hk = ? (altitude of the kite)
2) formula of sine ratio:
sin α = opposite-leg / adjacent leg
3) solution
sin α = x / s => x = sinα * s = sin(38°) * 50 ft = 30.78 ft
Hk = x + Ha = 30.78 ft + 4 ft = 34.78 ft => aption B. 34.8 feet
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