2D Geometry Questions and Answers

Theorem 111. Dilations preserve angle measure. Specifically, if noncollinear points A, B and C go to (necessarily) non-collinear points A', B' and C' under a dilation, then ∠ABC = ∠A'B'C'. [Draw a diagram and apply what you know.] The definition of a dilation says that the length of each segment coming from the center O is multiplied by the scaling factor k, that is, L(OP')=kL(OP). We would like to find out if the same relationship holds for every segment.

Given the points C(-1,-5) and D(-7, 11), find the coordinates of point E on CD such that the ratio of CE to ED is 5:3.

Find the equation of the hyperbola with vertices V(0, 2) and V'(4, 2) with one minor point at M(2, 3).

Two LORAN radio stations A and B are 190 kilometers apart and send a simultaneous radio signal to a ship. The signal from B arrives 0.0002 seconds before the signal from A. If the signal travels 300,000 kilometers per second, find the equation of the hyperbola on which the ship is positioned. Use a coordinate system where the origin is the midpoint between the stations, and the stations lie on the x-axis.

Two tangent lines at the endpoints of a given arc containing 120° are drawn, and a circle is inscribed into the figure bounded by these tangent lines and the arc. Prove that the circumference of this circle is equal to the arc length of the given arc.

Critical thinking. In ⊙ R, the measure of arc AB = 60°, the measure of arc BC 25°, the measure of arc CD = 70°, and the measure of arc DE 20°. Find two possible values for the measure of arc AE.

Use the information provided to answer Part A and Part B. Justify your answer using the CER method for part A as follows: Claim - The y - coordinate of the center of the circle is____ Evidence - Show work used to solve for/find the y coordinate. Reasoning - Use the work you showed in "Evidence" to justify your "Claim". Come up with your own CER response for part B. The equation x^2 + y^2 - 4x + 2y = b describes a circle. A)Determine the y-coordinate of the center of the circle. b)The radius of the circle is 7 units. What is the value of b in the equation?

The rectangle on the right is a scaled copy of the rectangle on the left, Identify the scale factor. Express your answer as a whole number or fraction in simplest form, first rectangle second rectangle l=14,b=6 l=56,b=24

The jar on Mrs. Wilson's desk contains 20 green paper clips, 30 red paper clips, 15 white paper clips, and 10 black paper clips. She selects a paper clip without looking, does not replace it, and selects another. Determine each probability. Show complete work on your worksheet!!! (a) P(both paper clips are white) = Fractional set up:________________ 2 decimal places:________________ (b). P(second paper clip is green if first paper clip is black) = Fractional set up:_______________ 2 decimal places:_______________ (c). P(second paper clip is green if first paper clip is green) = Fractional set up:_______________ 2 decimal places:_______________

32 Triangle ABC has vertices with coordinates A(-1,-1), B(4,0), and C(0,4). Prove that ABC is an isosceles triangle but not an equilateral triangle. [The use of the set of axes below is optional.]

Using words + diagrams as needed, describe how you would construct the length (3+√7)/2 using only compass and straightedge. It will be helpful to split this construction into a few steps.

How many triangles are there that have a = 7, b=5, and area 5√5. There is/are _____ triangle(s). (Type a whole number.)

The points (3, 1) and (6,5) are the endpoints of a radius of a circle, and the point (9, 2) is inside the circle. Write the standard equation of the circle. (x - _____)^2 + (y - _____)^2 = (______)^2

Construct a regular pentagon inscribed in a circle of radius 1+ √5. Compute the exact side length of the regular pentagon and the angles you get "for free". Then construct a rhombus with side length 3+√2 and an angle of measure 54°. Compute the exact lengths of the diagonals of the rhombus.

A hummingbird is flying toward a large tree that has a radius of 6 feet. When it is 33 feet from the center of the tree, its lines of sight form two tangents. What is the measure of the arc on the tree that the hummingbird can see? a 159.05° b 158.05° d 79.525° d 80.525°

Given that P(A and B) = 0.20 and P(AB) = 0.40, what must the P(B) = ? a)0.1 b)0.2 c)0.4 d)0.5

Which answer would make △ WTV- △RSP by the AA similarity postulate? 1. WT/RS= SP/TV= WV/RP 2. m angle W = m angle V and m angle R = m angle P 3. m angle T = m angle S and m angle V = m angle P 4. WT/RS= TV/SP= WV/RP

In the figure above, which of the following would allow you to prove triangle CBA similar to triangle EDA? 1) angle DEA = angle A 2) CB= EA 3) No more information is needed; they are already similar 4) DA= CB

If you roll a standard die one time, what is the probability that it will land on a 5? A) 3/6 B) 1/6 C) 4/6 D) 5/6

Suppose that while practice free throws, you made 26 shots and missed 14. Based on these results, determine the experimental probability that you will miss your next free throw. The experimental probability that you will miss your next free throw is ___________.

Suppose A and B are independent events. If P(A) = 0.65 and P(B)=0.20, determine the following : P(A and B) = _____________ P(A or B) = _____________

Which triangle is similar to an equilateral triangle with side lengths of 5.2 cm? Select one: 1. an equilateral triangle with perimeter 10 feet 2. any triangle with perimeter 15.6 cm 3. an isosceles triangle where two of the side lengths are 5.2 cm 4. a right triangle with perimeter 15.6 cm

What is the formula for the Area of a circle? A = 2*π*Radius A = π*Radius^2 of A = π*Radius*Radius A = π*Diameter A = Length * Width

What is the formula for Circumference? (there may be more than one right answer) C = π*Radius^2 or A = π*Radius*Radius C = 2*π*Radius C = Length * Width C = π*Diameter

2. The circumference is 24π and mCD = 60°. Length of CD =

Maurice is building a square garden and wants to put a diagonal divider to separate his vegetables from his herbs. He is trying to figure out how long the divider needs to be if his garden has a perimeter of 40 meters. How long is the diagonal divider? 10 meters 100 sq. meters 10√2 meters 40 meters

Six angles of a heptagon measure 107°, 139°, 131°, 110°, 145°, and 128°. What is the measure of the seventh angle? a. 900° b. 159° c. 140° d. 137⁰

A circle has an arc with measure 15° and length 88π. What is the diameter of the circle?

A quadrilateral has vertices A = (0,0), B = (2,4), C = (0,5), and D= (-2, 1). Select the most precise classification for quadrilateral ABCD. A. quadrilateral B. parallelogram C. rectangle D. square

A circle and line are drawn. How many intersection points are possible? Select all possible answers. 0 1 2 3 4

X is a normally distributed random variable with mean 37 and standard deviation 15. What is the probability that X is between 22 and 52? Use the 0.68 - 0.95 - 0.997 rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

Stacy rolls a pair of six-sided fair dice. The probability that the sum of the numbers rolled is either a multiple of 3 or an even number is ____, and two events are ______ exclusive.

In the diagram, mCD=128° mDA=76°. A. 104° B. 116° C.128° D. 102° E. 96°

In the diagram, rectangle ABCD is split in half by AC. What is the value of tan x? A.) 3/4 B.) 4/5 C.) 3/5 D.) 5/4 E.) 4/3

Select the correct answer. Your personal library contains books written by 165 authors. 60% of the authors are men. 40% of the authors write only nonfiction works. Also, 40 of the male authors write only nonfiction works. Assuming that your library has only one book by each author, what is the probability that a book picked at random is either a work written by an author who writes only nonfiction or a work written by a man? A.) 3/5 B.) 58/165 C.) 25/33 D.) 2/5

The radil of Earth and Pluto are 6,371 kilometers and 1,161 kilometers, respectively. Approximately how many spheres the size of Pluto does it take to have the same volume as Earth? A. 121 B. 154 C. 165 D. 176 E. 189

In a circle centered at point O, the ratio of the area of sector AB to the area of the circle is 3/5. What is the approximate measure in radians of the central angle corresponding to AB? Round the answer to two decimal places. A. 3.14 B. 3.35 C. 3.62 D. 3.77 E. 3.85

A circle has an area of 49π in². Which of the following is the circumference of the circle in terms of pi (π)? 7π in 28π in 14π in 3.5π in

Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s). Points A and B are the endpoints of an arc of a circle. Chords are drawn from the two endpoints to a third point, C, on the circle. Given mAB = 64° and ABC = 73° m ACB =____ and mAC =_____

While driving from his home to his workplace, Chris has to pass through two traffic lights. The general probability of getting a red signal at the first traffic light (event A) is 36%. The general probability of getting a red signal at the second traffic light (event B) is 25%. The probability of Chris getting a red signal at both traffic lights is 9%. In this scenario, A and B are ______ events.

Type the answer in the box. If necessarry, use /for the fraction bar. Give your answe in reduced form. A card is drawn at a random from a well-shuffled deck of playing cards. The probability that the card drawn is an ace or a red card is

Find all the complex roots. Leave your answer in polar form with the argument in degrees. The complex fifth roots of 3. Zo=(cos+isin) (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed. Type any angle measures in degrees.) z₁=(cos+isin) (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed. Type any angle measures in degrees.) z2=(cos+isin) (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed. Type any angle measures in degrees.) z3=(cos+ i sin) (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed. Type any angle measures in degrees.) z4=(cos+ i sinº) (Simplify your answer, including any radicals. Type an exact answer, using radicals as needed. Type any angle measures in degrees.)

Question) Identify what type of conic section, or if they are just straight lines, are the quadratic shapes below. Transform to the main axis. Express x¹ = [X₁ X₂] in terms of new coordinates y¹ = [y₁ Y₂]. a) 7x₁²+ 6x1x2 + 7x₂² = 200 b) 9x₁² - 6x1x2+ X₂² = 40

Select the correct answer.An experiment consists of rolling a six-sided die to select a number between 1 and 6 and drawing a card at random from a set of 10 cards numbered 1, 2, 3.... 10. Which event definition corresponds to exactly one outcome of the experiment? A. Both numbers are 5s. B. Neither number is a 5. C.The sum of the numbers is 3. D. One of the numbers is a 3.

If the endpoints of AB have the coordinates A(9, 8) and B(-1, -2), what is the midpoint of AB? A. (5,3) B. (4,5) C. (5,5) D. (4,3)

Equations of Circles When center is (0, 0): x² + y² = r² when center is not (0, 0): (x - h)² + (y - k)² = r² Write the equation of a circle with center (0, 0) and a radius of 15.

A circle is centered at the point (-3, 2) and passes through the point (1,5). The radius of the circle is ___ units. The point (-7,__) lies on this circle.

What is the radius of the circle x² + y² + 16 = 9x?

What is the center of the circle x² + y² - 36 = 16x? Simplify any fractions. _____ , ______

The point Q lies on the segment PR. Find the coordinates of Q so that PQ is 3/5 of PR Coordinates of Q:(______)