Geometry and Trigonometry
Which of the following pieces of information is sufficient to prove that triangle ABC is an isosceles triangle?
i. AB is congruent to BC
ii. Angle A is congruent to angle C
a) i is sufficient, but ii is not.
b) ii is sufficient, but i is not.
c) Either i or ii is sufficient
d) Neither i nor ii is sufficient
Answer: c) Either i or ii is sufficient
(x -3)2 + (y + 16)2 = 289
The graph of the given equation is a circle in the xy-plane. The point (q,s) lies on the circle. Which of the following is NOT a possible value of q?
A. -17
B. -3
C. 3
D. 17
Answer: A. -17
3x2 - 7x -1 = 0
What are the solutions to the given equation?
A. x = 7 (+-) √37 / 6
B. x = 7 (+-) √61 / 6
C. x = -7 (+-) √37 / 6
D. x = -7 (+-) √61 / 6
Answer: B. x = 7 (+-) √61 / 6
If p = 3x + 4 and v = x + 5, which of the following is equivalent to pv - 2p +v?
A. 3x2 + 12x + 7
B. 3x2 + 14x + 17
C. 3x2 + 19x + 20
D. 3x2 + 26x + 33
Answer: B. 3x2 + 14x + 17
2(8x) + 4(7y) = 12
-2(8x) + 4(7y) = 12
The solution to the given system of equations is (x,y). What is the value of 8x + 7y?
Answer: 3
Triangle FGH is similar to triangle JKL, where angle F corresponds to angle J and angles G and K are right angles. If sin(F) = 308/317, what is the value of sin (J)?
A. 75/317
B. 308/317
C. 317/308
D. 317/75
Answer: B. 308/317
Because the triangles are similar, this means the measure of angle F equals the measure of angle J. If two angles are equal then their sine values msut be equal.
(x + 4)2 + (y - 19)2 = 121
The graph of the given equation is a circle in the xy-plane. The point (a,b) lies on the circle. Which of the following is a possible value for a?
A. -16
B. -14
C. 11
D. 19
Answer: B. -14
x2 + 2x + 1 = 4
What is the positive solution to the given equation?
Answer: 1
The function h is defined by h(x) = ax + b, where a and b are positive constants. The graph of y = h(x) in the xy-plane passes through the points (0,10) and (2,13). What is the value of ab?
A. 13
B. 18
C. 20
D. 26
Answer: B 18
f(x) = 4x + b
For the linear function f, b is a constant and f(7) = 28. What is the value of b?
A. 0
B. 1
C. 4
D. 7
Answer: A. 0
In triangle XYZ, angle Z is a right angle and the length of YZ is 24 units. If tan X = 12/35, what is the perimeter, in units, of triangle XYZ?
A. 188
B. 168
C. 84
D. 71
Answer: B. 168
tan (X) = Opposite/ Adjacent
12/35 = 24/XZ --> Cross multiply --> 12(XZ) = 24 * 35 --> 12(XZ) = 840
--> XZ = 840/12 --> XZ = 70
Find the Hypotenuse
702 + 242 = c2 --> 4,900 + 576 = c2 --> 5,476 = c2 --> c =sqrt(5,476) = 74
so, XY = 74
Perimeter = 74 + 70 + 24 = 168
x2 + y2 + 6x + 6y = -45/4
The equation of a circle in the xy-plane is shown. What is the radius of the circle?
Answer: 2.5 or 2
Method 1 : Put eqaution in desmos. Find the diameter and then the radius.
Method 2:
If ax -3 is a factor of 6x3 + 27x2 - 54x, where a is a positive constant, what is the value of a?
Answer: 2
Put the equation in Desmos.
For ax-3 to be a factor of the polynomial, it must be equal to 0; hence :
ax - 3 = 0 --> x = 3/a
Substitute x = 3/a into the equation in Desmos and equate the entire equation to 0 using the regression sign (~).
The answer is 2
The product of two positive integers is 462. If the first integer is 5 greater than twice the second integer, what is the smaller of the two integers?
Answer: 14
2x - 3y = 5
One of the two equations in a system is given. The system has an infinite number of solutions. Which equation could be the other equation in the system?
A. 4x - 6y = 10
B. 4x + 6y = 10
C. 2x - 3y = 10
D.2x + 3y = 10
Answer: A. 4x - 6y = 10
For two acute angles, angle Q and angle R, cos(Q) = sin(R). The measures, in degrees, of angle Q and angle R are x + 52 and 4x +13, respectively. What is the value of x?
A. 5
B. 13
C. 23
D. 38
Answer: A. 5
Rule: sin (angle) = cos (90 - angle) [vice versa] --> this means the angles are complementary (adds up to 90)
Angle Q + angle R = 90
(x + 52) + (4x +13) = 90 (can put this in desmos) to get 5
Or
x + 4x + 13 + 52 = 90 --> 5x + 65 = 90 --> 5x = 90 - 65 --> 5x =25 --> x =25/5 --> x=5
x2 + x√t + y2 - y - 65 = 0
The given equation, where t is a constant, defines a circle in the xy-plane. The radius of this circle is √83. What is the value of t?
A. 73
B. 71
C. 37
D. 35
Answer: B
h(x) = 2(x - 4)2 - 32
The quadratic function h is defined as shown. In the xy-plane, the graph of y = h(x) intersects the x-axis at (0, 0) and (t, 0), where t is a constant. What is the value of t?
A. 1
B. 2
C. 4
D. 8
Answer: D
Put the equation in Desmos and look at where the line crosses the x-axis
f(x) = x2- 48x + 2,304
What is the minimum value of the given function?
Answer: 1728
3x = 36y - 45
One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?
A. x = 4y
B. 1/3x = 4y
C. x = 12y -15
D. 1/3x = 12y -15
Answer: B. 1/3x = 4y
In triangle ABC, the measure of angle A is 52° and AC = 30. In triangle PQR, the measure of angle P is 52° and PR =120. Which additional piece of information is sufficient to prove that triangle ABC is similar to triangle PQR?
A. AB =50 and PQ = 50.
B. AB = 50 and QR = 200.
C. The measures of angle B and angle R are 32° and 96°, respectively.
D. The measures of angle B and angle Q are 52° and 32°, respectively
Answer: C. The measures of angle B and angle R are 32° and 96°, respectively.
To prove similarity, we only need to show that a second pair of angles matched (all angles in triangle adds up to 180°).
We know: Angle A = Angle P = 52°
Given: Angle B = 32° and Angle R = 96°
Find missing angle in ABC: 180 - (52 + 32) = 96° (angle C)
Find missing angle in PQR: 180 - (52 +96) = 32° (angle Q)
Result: Both triangles have identical angles, so they are similar.
A circle in the xy-plane has its centre at (-1,1). Line t is tangent to this circle at the point (5, -4). Which of the following points also lies on line t?
A. (0,6/5)
B. (4,7)
C. (10, 2)
D. (11,1)
Answer: C
Find slope of the radius: y2 - y1/x2 - x1 --> -4 - 1/5 - (-1) = -5/6
Find the slope of line t: The tangent line is perpendicular to the radius at the point of tangency. Perpendicular slopes are negative reciprocals. So, t = 5/6
In Desmos, use the known point (5, 4) and the answer choices to see which point lies on the line.
A quadratic function can be used to model the height, in feet, of an object above the ground in terms of the time, in seconds, after the object was launched. According to the model, an object was launched into the air from a height of 784 feet 7 seconds after it was launched. Based on the model, what was the height, in feet, of the object 3 seconds after it was launched?
Answer: 528
One solution to the given equation can be written as 1 + √k, where k is a constant. What is the value of k?
A. 8
B. 10
C. 20
D. 40
Answer: B. 10
The graph of the equation ax +ky = 6 is a line in the xy-plane, where a and k are constants. If the line contains the points (-2,-6) and (0,-3), what is the value of k?
A. -2
B. -1
C. 2
D.3
Answer: A. -2