Geometry: MIXED

The line represented by the equation 6x − 2y = 14 is transformed by a dilation centered at (2,-1) with a scale factor of 2. The image of the line is…

the original line

In the diagram below, ERM~JTM. Which statement is not always true?

CosT = RM/EM

Which of the following ratios is equivalent to sin D?

Sin D = FE/ED

In right triangle JOE, hypotenuse JO = 31.8 and m∠J = 38°. To the nearest tenth, determine the length of EJ.

25.1

X = ___

x=2.8

The Washington Monument, shown below, is in Washington, D.C. At a point on the ground 200 feet from the center of the base of the monument, the angle of elevation to the top of the monument is 70.19°.

What is the height of the monument, to the nearest foot?

555 feet

An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 24 feet up. The ladder makes an angle of 73∘ with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary.

25.1 feet

From the observation deck of a skyscraper, Isabella measures a 45∘ angle of depression to a ship in the harbor below. If the observation deck is 1036 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.

1036 feet

The ramp shown in the diagram below has an angle of elevation of 4.8º. The ramp is built to a landing 0.6 m above the ground.

Determine and state the length of the ramp, to the nearest tenth of a meter.

7.2

A boat heading out to sea starts out at Point A, at a horizontal distance of 649 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light to be 5∘. At some later time, the crew measures the angle of elevation from point B to be 2∘. Find the distance from point A to point B. Round your answer to the nearest foot if necessary.

The boat traveled 977 feet from point A to point B.

Cape Canaveral, Florida is where NASA launches rockets into space. As modeled in the diagram below, a person views the launch of a rocket from observation area A, 3280 feet away from launch pad B. After launch, the rocket was sighted at C with an angle of elevation of 15°. The rocket was later sighted at D with an angle of elevation of 31°.

Determine and state, to the nearest foot, the distance the rocket traveled between the two sightings, C and D.

1092

In a right triangle, the acute angles have the relationship sin(3x − 7)° = cos(x + 1)°.
Determine and state the value of the larger acute angle.

65

In right triangle ABC with the right angle at C, sinA = 2x + 0.1 and cosB = 4x − 0.7. Determine and state the value of x.

0.4

(Remember since we’re not dealing with degrees A + B = 90 doesn’t apply. Instead its A = B)

In a right triangle, cos (9x)∘ = sin (6x−10)∘. Find the larger of the triangle’s two acute angles.

60

In right triangle ABC, m∠A=90∘, sin B=3x+0.04, and cos C=8x−0.41. Solve for x.

x=0.09

In a right triangle, sin (7x−2)∘ = cos (x−10)∘. Solve for x. Round your answer to the nearest hundredth if necessary.

12.75

An equilateral triangle is continuously rotated around one of its altitudes.  The three-dimensional object formed is a

1)	cone	3)	cylinder
2)	sphere	4)	pyramid

(Multiple choice)

1)

cone

1)	19.6	3)	40.4
2)	25.1	4)	51.7

2)

25.1

To the nearest cubic centimeter, the volume of the hemisphere is

1)	201	3)	1072
2)	268	4)	2145

201

3)

1072

1)	24	3)	12
2)	18	4)	10

2)

18

A wooden toy block can be modeled by a pyramid with a square base, as shown below.  The height of the block is 17.4 cm and the square base has a side length of 8.2 cm.

The block is made of solid oak, which has a density of 0.77 g/cm³.  What is the mass of the block, to the nearest gram?

1)	300	3)	637
2)	506	4)	901

1)

300

1)	28	3)	7
2)	14	4)	4

2)

14

1)	5	3)	10
2)	5√2	4)	10√2

3)

10

What is the area of RST , to the nearest square centimeter?

1)	59	3)	29
2)	36	4)	21

3)

29

What are the coordinates of the center of this dilation?

(2,1)

Which equation is always true?

4. PE/FE = PG/FT

A section of sidewalk in the shape of a rectangular prism is being replaced.  The sidewalk is 10 feet long, 4 feet wide, and 4 inches deep.  A brand of concrete mix yields 0.6 cubic foot of concrete per bag.  What is the minimum number of bags of concrete mix that must be purchased to completely replace this sidewalk?

1)	22	3)	26
2)	23	4)	27

2)

23

Which equation represents the image of the line after this dilation?

1) y = 2/3x - 12

1)	9	3)	15
2)	12	4)	16

2)

12

Square ABCD has an area of 36.  If the square is dilated by a scale factor of 1/2 centered at A, what is the area of its image?

1)	9	3)	72
2)	18	4)	144

1)

9

A solid metal cube has an edge length of 5 cm and a mass of 982.5 grams.  Using the table above, determine and state the type of metal from which this cube is made.

IRON

Determine and state the length of the ramp, to the nearest tenth of a meter.

Answer: 7.82

Joan wants to fill an empty 75-liter fish tank with water.  She uses a cylindrical bucket with a diameter of 20 cm.  Determine and state the maximum number of buckets of water, filled to an exact height of 26 cm, Joan can put into the fish tank before it overflows.  [1000 cm³=1 liter]

9 buckets of water

As modeled in the diagram below, two cables are attached from a point on a tree 12 feet above the ground.  The longer cable is anchored on the ground 3 feet farther from the tree than the shorter cable is anchored.  The angle of elevation between the shorter cable and the ground is 50º.

Determine and state, to the nearest foot, the distance from the base of the tree to the point where the longer cable is attached to the ground.  Determine and state, to the nearest degree, the angle of elevation between the longer cable and the ground.

13, 43

What are the three main methods used to prove two triangles are similar?

SSS, SAS, and AA are the three main methods used to prove triangle similarity. SSS requires all three pairs of corresponding sides to be proportional. SAS requires two pairs of sides to be proportional and the included angle to be congruent. AA requires two pairs of corresponding angles to be congruent.

What does it mean for two triangles to be similar?

Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. This means they have the same shape but not necessarily the same size. The scale factor describes how much one triangle is enlarged or reduced compared to the other.

What symbol represents similarity in geometry?

The symbol ~ represents similarity between two figures. It shows that the shapes have the same form but different sizes. Corresponding angles are equal and corresponding sides are proportional.

What must be true about the sides of similar triangles?

The sides of similar triangles must be proportional. This means each pair of corresponding sides has the same ratio. This constant ratio is called the scale factor.

What must be true about the angles of similar triangles?

The angles of similar triangles must be congruent. This means each pair of corresponding angles has the same measure. This is what ensures the triangles have the same shape.

What is SOH-CAH-TOA used for?

SOH-CAH-TOA is used to relate angles and side lengths in right triangles. It helps find missing side lengths or angle measures. Each part represents a trigonometric ratio.

What does SOH stand for?

SOH stands for sine equals opposite over hypotenuse. It relates the opposite side and hypotenuse of a right triangle. It is used when those sides and an angle are involved.

What does CAH stand for?

CAH stands for cosine equals adjacent over hypotenuse. It relates the adjacent side and hypotenuse. It is useful when working with those sides and an angle.

What does TOA stand for?

TOA stands for tangent equals opposite over adjacent. It relates the opposite and adjacent sides of a right triangle. It is helpful when those two sides are known or needed.

When do you use inverse trig functions like sin^-1, cos^-1, and tan^-1?

Inverse trig functions are used to find angle measures. They are applied when the side lengths of a right triangle are known. Each inverse function corresponds to its regular trig ratio.

What is the formula for the volume of a cone?

The volume of a cone is V = (1/3)πr^2h. It uses the radius of the base and the height. The formula shows the cone has one-third the volume of a cylinder with the same base and height.

What is the formula for the volume of a cylinder?

The volume of a cylinder is V = πr^2h. It is found by multiplying the area of the circular base by the height. This formula applies to all cylinders.

What is the formula for the volume of a pyramid?

The volume of a pyramid is V = (1/3)Bh. The base area is multiplied by the height and then divided by three. This applies to all pyramid shapes.

What is the formula for the volume of a sphere?

The volume of a sphere is V = (4/3)πr^3. It depends only on the radius of the sphere. The radius is cubed because volume is three-dimensional.

What is the formula for the volume of a prism?

The volume of a prism is V = Bh. The base area is multiplied by the height. This works for any type of prism.

How do you find the radius if you are given the diameter?

The radius is found by dividing the diameter by 2. This is because the diameter is twice the radius. This relationship applies to all circles.

What is the formula for circumference?

The circumference of a circle is C = 2πr. It represents the distance around the circle. It can also be written as C = πd.

What is the density formula?

Density is equal to mass divided by volume. It measures how much mass is in a given space. Common units include grams per cubic centimeter.

How do you find volume using density?

Volume is found by dividing mass by density. This comes from rearranging the density formula. It is useful when density and mass are known.

How do you find mass using density?

Mass is found by multiplying density by volume. This is another rearrangement of the density formula. It helps when volume and density are given.

What happens to lines after a dilation?

After a dilation, the image of a line remains parallel to the original line. This means they have the same slope. The size may change, but the direction stays the same.

What happens if the center of dilation is on the original line?

If the center of dilation is on the line, the image will have the same slope and the same y-intercept. This means the line stays exactly in the same position. Only the scale changes.

What is the formula for scale factor?

The scale factor is equal to image divided by preimage. It describes how much a figure is enlarged or reduced. A value greater than 1 means enlargement, and less than 1 means reduction.

What is the area formula for a rectangle or square?

The area is length times width. This applies to both rectangles and squares. In a square, both values are the same.

What is the area formula for a right triangle?

The area is (1/2) times base times height. The base and height must be perpendicular. This formula is specific to triangles.

What is the area formula using sine?

The area is (1/2)ab sin(C). It uses two sides and the included angle. This is useful when height is not directly given.

How do you find the length of a line on a coordinate plane?

You find the rise and run between two points. Then use the Pythagorean theorem. This gives the distance between the points.

What is the Pythagorean theorem?

The Pythagorean theorem is a^2 + b^2 = c^2. It applies to right triangles. The hypotenuse is always the longest side.

What defines an isosceles triangle?

An isosceles triangle has two equal sides. The angles opposite those sides are also equal. This creates symmetry in the triangle.

What must the angles in any triangle add up to?

The angles in any triangle must add up to 180 degrees. This is always true in Euclidean geometry. It helps solve for missing angles.

How does perimeter change after a dilation?

The perimeter of the image equals the original perimeter times the scale factor. This is because all lengths are scaled equally. It changes linearly.

How does area change after a dilation?

The area of the image equals the original area times the scale factor squared. This is because area is two-dimensional. Changes grow faster than perimeter.

What does the symbol ≅ mean?

The symbol ≅ means congruent. It shows that two figures have the same shape and size. All corresponding sides and angles are equal.

What does ABC ≅ DEF mean?

It means triangle ABC is congruent to triangle DEF. All corresponding sides and angles match exactly. The order shows which parts correspond.

What is a horizontal stretch?

A horizontal stretch multiplies the x-values by a constant. This changes the width of a graph. The graph becomes wider or narrower.

What is a vertical stretch?

A vertical stretch multiplies the y-values by a constant. This changes the height of a graph. The graph becomes taller or shorter.

What are rigid motions?

Rigid motions include translations, rotations, and reflections. They do not change the size or shape of a figure. Distances and angles remain the same.

What happens in a translation?

A translation slides a figure from one place to another. All points move the same distance and direction. The shape and size stay the same.

What happens in a rotation?

A rotation turns a figure around a fixed point. The angle of rotation determines how far it turns. The figure keeps its shape and size.

What happens in a reflection?

A reflection flips a figure over a line. This line is called the line of reflection. The image is a mirror image of the original.

If sin(A) = cos(B), what relationship exists between A and B?

A + B = 90 degrees. This means the angles are complementary. This comes from trig identities.

What is the area of a sector formula?

The area of a sector is (theta/360) times πr^2. It represents part of a circle. The angle determines how large the sector is.

What is the arc length formula?

The arc length is (theta/360) times 2πr. It measures the distance along the circle. The angle controls the portion of the circle.

What is true about diagonals in a rectangle?

The diagonals in a rectangle are congruent. This means they are equal in length. They also bisect each other.

What is true about diagonals in a rhombus?

The diagonals in a rhombus are perpendicular. This means they intersect at right angles. They also bisect each other.

What is true about diagonals in a square?

The diagonals in a square are both congruent and perpendicular. This combines properties of rectangles and rhombuses. They also bisect each other.

What is true about diagonals in a parallelogram?

The diagonals bisect each other. This means they cut each other in half. They are not necessarily equal or perpendicular.

What must the angles in a quadrilateral add up to?

The angles in any quadrilateral must add up to 360 degrees. This is true for all four-sided shapes. It helps find missing angles.

What is true about opposite sides in a parallelogram?

Opposite sides are parallel and congruent. This means they never meet and are equal in length. This defines the shape.

What is true about diagonals in an isosceles trapezoid?

The diagonals are congruent. This means they have the same length. This is a special property of this trapezoid.

What is true about all sides of a rhombus?

All sides of a rhombus are congruent. This means every side has the same length. It is like a tilted square without right angles.

A shipping container is in the form of a right rectangular prism, with dimensions of 35 ft by 8 ft by 9 ft 9 in. If the container holds 1420 cubic feet of shipped goods, what percent is full? Round your answer to the nearest whole number if necessary.

52% full

Triangle XYZ, with vertices X(-8,3), Y(-3,7), and Z(-9,8), is drawn on the coordinate grid below.

What is the area, in square units, of triangle XYZ?

x = __

x ≈ 3.2

The regular polygon below is to be rotated about its center. Which angle of rotation would carry the figure onto itself?

A. 210°

B.135°

C. 270°

D. 324°

D. 324°

Solve for x. Round your answer to the nearest tenth if necessary.

x = 5

If a tree has a height of 165 feet, what would be the angle of elevation from level ground measured from 95 feet away? Round your answer to the nearest tenth of a degree.

60.1 º

In a right triangle, cos (7x)∘ = sin (2x−8)∘. Solve for x. Round your answer to the nearest hundredth if necessary.

x=10.89

A toy ball can be modeled as a sphere. Brooklyn measures its circumference as 65.8 cm. Find the ball’s volume in cubic centimeters. Round your answer to the nearest tenth if necessary.