Mathematics College

## Answers

**Answer 1**

**Answer:**

m∠14 = 138°

**Step-by-step explanation:**

∠2 = ∠5 = 63° (vertically opposite angles)

∠9 +∠8 = 180° (linear pair)

=> 105° + ∠8 = 180

=> ∠8 = 180° - 105° = 75°

∠5 + ∠8 + ∠11 = 180° (sum of angles in a triangle is 180)

=> 63° + 75° + ∠11 = 180°

=> 138° + ∠11 = 180°

=> ∠11 = 180° - 138° = 42°

∠11 + ∠14 = 180° (linear pair)

=> 42° **+ ∠**14 = 180°

=> ∠14 = 180° - 42° = 138°

Hope you understood!!

## Related Questions

Joe went to the hobby shop and bought 2 model sports cars and spent$8.95 each on some paints. If he spent a total of $23.65, what was the costof each model car?*

### Answers

Step 1

Find the cost of the 2 model cars

Total cost of two model cars and paint = $23.65

Cost of some paint = $8.95

Cost of 2 model cars = $23.65 - $8.95

= $14.7

Step 2

Calculate cost of each model car

Cost of each model car = $14.7/2

= $7.35

10 points5) Which value is NOT located between these two numbers on thenumber line?V100527 - 17V5

### Answers

The values on the Number line are

[tex]\frac{\sqrt{100}}{5}=\frac{10}{5}=2[/tex][tex]2\pi^2-17=2(3.14)^2-17=2.73[/tex]

So the values in the number line are between 2 and 2.73

Option 1: √5≅2.24

Option 2:π is a mathematical constant, its first digits are 3.14

The number NOT included between the value son the number line is pi (option 2)

using the distributive property simplify the expression to determine the cost 5(18+3x)= +

### Answers

5(18+3x)= ( 5 x 18 ) + ( 5 X 3x )

A rectangle is placed around a semi circle as shown below the length of the rectangle is 12mm. Find The area of the shaded region. Use the value 3.14 for pi, and do not round your answer be sure to include the correct unit in your answer.

### Answers

**Explanation **

We are given a rectangle placed around a semi-circle as shw in the image below:

We are required to determine the area of the shaded portion.

**This is achieved thus:**

The area of the shaded portion is given as:

[tex]\begin{gathered} Area=Area\text{ }of\text{ }rectangle-Area\text{ }of\text{ }semicircle \\ Area=A_r-A_s \end{gathered}[/tex]

The dimension of the rectangle is given as:

[tex]\begin{gathered} Length=Diameter\text{ }of\text{ }the\text{ }semicircle=12mm \\ Width=Radius\text{ }of\text{ }the\text{ }semicircle=\frac{12}{2}=6mm \end{gathered}[/tex]

Therefore, the area of the shaded portion is:

[tex]\begin{gathered} Area=A_r-A_s \\ Area=(l\times w)-(\frac{\pi r^2}{2}) \\ Area=(12\times6)-(\frac{3.14\times6^2}{2}) \\ Area=(72)-(56.52) \\ Area=15.48mm^2 \end{gathered}[/tex]

**Hence, the answer is:**

[tex]Area=15.48mm^{2}[/tex]

the manger of the video department at a department store plans to purchase a large number of DVDs of a recent movie. one supplier is selling boxes of 25 DVD movies for $230 and a second supplier is selling boxes of 12DVD movies for $170. only complete boxes of DVD movies can be purchased.a.)if the managercan purchase boxes of DVD moviesfrom either ir bothsuppliers, determine the maximu mnumber of dvd movies that can be purchased for $415 indicate how manyboxes of 25 and how many boxes of 12 will be purchased.b) how much will it cost?

### Answers

If the manager has only $415, he must try to get the deal in which he gets the greatest amount of DVDs for the leats price, which is the 25 DVDs for 230 because the unit price per DVD is less than in the other deal.

However if he gets that deal

[tex]415-230=185[/tex]

he still has enough money to get 1 box of DVDs from the other deal

[tex]185-170=15[/tex]

According to this the manager can get 1 box of 25 DVDs and 1 box of 12 DVDs.

It is gointgo cost him $400 for the buying of the DVDs.

[tex]230+170=400[/tex]

Find the distance between the two points rounding to the nearest tenth (if necessary). (-6,6) and (3, -6)

### Answers

**The formula used for calculating distance between two points is expressed as shown**:

[tex]D\text{ = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Given the coordinates (-6,6) and (3, -6)

x1 = -6, y1 = 6, x2 = 3 and y2 = -6

**Substitute this values into the formula above as shown:**

[tex]\begin{gathered} D\text{ = }\sqrt[]{(3-(-6))^2\text{ }+\mleft(-6-6\mright)^2} \\ D\text{ = }\sqrt[]{(3+6)^2\text{ }+(-12)^2} \\ D\text{ = }\sqrt[]{81\text{ }+144^{}} \\ D\text{ = }\sqrt[]{225} \\ D\text{ = }15 \end{gathered}[/tex]

**Hence the distance between the two points is 15 units**

Calculate the final price. Round all answers to the hundredths place and make sure to write your answer in the form of $12.34. Dinner: $32.50. Discount: 20%

### Answers

If the dinner has the 20% of discount. We only need to pay the 80% of the total price, so we get:

[tex]32.5\cdot\frac{80}{100}=26.00[/tex]

so the answer is

$26.00

y(s)= int 0 ^ e^ 4 cos z^ 0 z^ 2 dx then; y^ prime (s)=

### Answers

Given the following integral:

[tex]y(s)=\int\frac{cos\text{ }z^9}{z^2}dz[/tex]

We will find y'(s)

As we know the integral is the inverse of the differentiation

So, the first derivative of the integral can be obtained by removing the integral sign

So, the answer will be:

[tex]y^{\prime}(s)=\frac{cos\text{ }z^9}{z^2}[/tex]

Kent drives 270 miles in the same time that it takes Dave to drive 250 miles. If Kent averages 4 miles per hour faster than Dave, find their rates.

### Answers

Step1: Derive the speed equations from the data.

Given data are as follows

For Dave

Distance traveled = 250 miles

Let the speed of Dave be d miles per hour

For Kent

Distance traveled = 270 miles

Since Kent's speed is 4 miles per hour faster than Dave, we can represent this mathematically

so that If we represent Kent's speed by k. Then

k = 4 + d.

Distance traveled is given by the formula

Distance = Speed x Time

Therefore

[tex]\text{Time}=\frac{Distance}{\text{Speed}}[/tex]

So the time spent by Kent is

[tex]\text{time=}\frac{270}{d\text{ +4}}[/tex]

The time spent by Dave is

[tex]\text{time}=\frac{250}{d}[/tex]

Step 2: Since they both spend the same time, we will equate their time spent

So

[tex]\frac{270}{d+4}=\frac{250}{d}[/tex]

Step3: The next step is to solve the above equation

d (270) = 250 (d+4)

Expand the parenthesis

d x 270 = 250 x d + 250 x 4

270d = 250d + 1000

Collect like terms

270d - 250d = 1000

20d = 1000

Divide both sides by 20

d = 1000/20 = 50

d = 50 miles per hour

So Dave's rate is 50 miles per hour

Since we have been told that Kent travels 4 miles per hour faster than Dave, then

Kent = Dave + 4

K = 50 + 4

k= 54 miles per hour

Hence the rates are

**Kent = 54 miles per hour**

**Dave = 50 miles per hour**

Triangle TMQ is shown with vertices T (4, 1), M(1, 3), and Q(2, 3).Triangle TMQ is reflected across the line y = 1 to form triangle T'M'Q' what are the coordinates of Q'?

### Answers

In order to find the coordinates of the vertex T', M' and Q', use the following transformation:

**T(x,y) => T'(x , -(y - 1) + 1)**

Then, for the given points T, M and Q, you obtain:

T(-4 , 1) => T'(-4, -(1 - 1) + 1) = **T'(4 , 1)**

M(-1 , 3) => M'(-1, -(3 - 1) + 1) = **M'(1 , -1)**

Q(2 , -3) => Q'(2 , -(-3 -1) + 1) = **Q'(2 , 5)**

**Hence, the coordinates ofthe point Q' are (2 , 5)**

Consinder this set of expressions.3 -4. -2 + -3. -5 - 4part a: Simplfy each expression.A: 3 =B: -4 =C: -2 + -3 =D: -5 - -4=part b: graph each value on the number line. label the points.

### Answers

The absolute value of a negative number is the positive of that same number. Therefore,

**Part A**

[tex]\lvert3\rvert=3[/tex][tex]-\lvert4\rvert=-4[/tex][tex]\lvert-2\rvert+\lvert-3\rvert=2+3=5[/tex][tex]\lvert-5\rvert-\lvert4\rvert=5-4=1[/tex]

**part B**

Find the equation of the line that is parallel to y=4x+1 and contains the point (1,1)

### Answers

SOLUTION

The given equation is:

[tex]y=4x+1[/tex]

Notice that the slope of the given equation is **4**.

Recall the slopes of parallel line are equal.

Therefore the slope of the required line is **4**.

Since the slope of the required line is 4 and it is given that the line contains the point **(1,1)**

Then the equation of the line using Point-Slope Form is:

[tex]\begin{gathered} y-1=4(x-1) \\ y-1=4x-4 \\ y=4x-4+1 \\ y=4x-3 \end{gathered}[/tex]

Therefore the required equation is:

[tex]y=4x-3[/tex]

Without using technology, describe the end behavior of f(x) = 3x32 + 8x2 − 22x + 43.

### Answers

End behavior of a polynomial

In order to find the end behavior of a polynomial we simply must observe the higher exponential behavior. Since it is so much higher than the others terms it will indicate the end behavior of the total function.

In the case:

[tex]f(x)=3x^{32}+8x^2-22x+43.[/tex]

It is enough to analyze the end behavior of 3x³² in order to find the end behavior of the whole polynomial.

When x tends to infinity

When x tends to infinity

x ⇒ ∞

then

3x³² grows and grows (infinitely!)

3x³² ⇒ ∞

When x tends to minus infinity

When x tends to minus infinity

x ⇒ -∞

x takes negative numbers however x³² is always positive, because it has an even exponent, then

when x ⇒ -∞

then

3x³² grows and grows (infinitely too)

3x³² ⇒ ∞

**Answer- as x ⇒ -∞, 3x³² ⇒ ∞ and as x ⇒ ∞, 3x³² ⇒ ∞**

р2.345 67 8Adding one acute angle with one obtuse angle it should equal 180°TrueFalse

### Answers

Notice that we have a case of two parallel lines intersected by a transverse line that forms with them acute and obtuse angles.

It is absolutely true that all acute angles formed are of the SAME measure, and that all obtuse angles formed are of the SAME measure.

And given that one obtuse angle plus an acute angle in the picture render a straight line, it is the same as saying that their addition renders 180 degrees.

It is** TRUE** that adding any obtuse angle plus any acute angle in the image provided will render 180 degrees.

(0,2) and(8,0) Write an equation in standard form of the line that passes through the given points

### Answers

To find the equation of a line that pasess through two ponit, we'll first have to calculate the slope, using the formula:

[tex]m=\frac{y_2-y^{}_1}{x_2-x_1}[/tex]

Therefore,

[tex]m=\frac{0-2}{8-0}\rightarrow m=\frac{-2}{8}\rightarrow m=-\frac{1}{4}[/tex]

Then, we use the point-slope form equa with the slope we just found and the point (8,0)

[tex]\begin{gathered} y-0=-\frac{1}{4}(x-8)\rightarrow y=-\frac{1}{4}x+2\rightarrow y-2=-\frac{1}{4}x \\ \rightarrow-4y+8=x\rightarrow8=x+4y \\ \text{Therefore, the equation of the line is:} \\ x+4y=8 \end{gathered}[/tex]

A camera is originally $150 The store gives a discount and the camera is now priced at $112.50 Write the percentage discount for the camera

### Answers

We are asked to determine the percentage of discount if the price for an item goes from $150 to $112.50. To do that we will use the following relationship:

[tex]150-\frac{150x}{100}=112.5[/tex]

Where "x" is the percentage of the discount.

Now we solve for "x" first by subtracting 150 from both sides:

[tex]-\frac{150x}{100}=112.5-150[/tex]

Solving the operations:

[tex]-\frac{150x}{100}=-37.5[/tex]

Now we multiply both sides by 100:

[tex]\begin{gathered} -150x=-37.5\times100 \\ -150x=-3750 \end{gathered}[/tex]

Now we divide both sides by -150:

[tex]x=-\frac{3750}{-150}[/tex]

Solving the operations:

[tex]x=25[/tex]

Therefore, the percentage of the discount is 25%.

I’m doing math homework for summer school and I don’t get this . I provided a picture so you can see the problem. Thanks so much -Chris

### Answers

**Solution:**

Given the table;

Thus;

[tex]f(-2)=1[/tex]

Find the slope of the line that is parallel to the line that passes through the following pair of points: (4, -2) and (16,0)

### Answers

**Answer**

**Slope = (1/6) = 0.1667**

**Explanation**

Twol parallel lines normally have their slopes equal to each other.

So, to find the required slope, we can just find the slope of the line that is parallel to it.

For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are **(x₁, y₁)** and **(x₂, y₂)**, the slope is given as

[tex]Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{y_2-y_1}{x_2-x_1}[/tex]

For this question,

**(x₁, y₁)** and **(x₂, y₂) **are **(4, -2) **and **(16, 0)**

**x₁ = 4**

**y₁ = -2**

**x₂ = 16**

**y₂ = 0**

[tex]\text{Slope = }\frac{0-(-2)}{16-4}=\frac{0+2}{12}=\frac{2}{12}=\frac{1}{6}=0.1667[/tex]

**Hope this Helps!!!**

erome is planning on tiling his bathroom floor with square tiles measuring 8 inches on each side. The bathroom measures 13 feet by 8 feet.a) What is the area of the floor, in square feet? _________square feetb) Convert this area to square inches.______ square inchesc) How many 8 inch square tiles will Jerome need to cover the floor? _____tiles

### Answers

**Given**

Side of Square tiles = 8 inches

Length of bathroom = 13 feet

Breadth of bathroom = 8 feet

**Find**

**a) **area of the floor

b) Convert this area to square inches.

c) Number of tiles to cover the floor

**Explanation**

a) Area of floor is given by

[tex]\begin{gathered} l\times b \\ 13\times8 \\ 104ft^2 \end{gathered}[/tex]

b) as we know ,

1 square foot = 144 square inches .

so , 104 square feet = 144 * 104 = 14976 square inches

c) to find the number of tiles ,

we first find the area of 1 tile

area of tile = side * side = 8 * 8 = 64 square inches

so , number of tiles = area of floor / area of 1 tile.

hence ,

[tex]\begin{gathered} \frac{14976}{64} \\ 234 \end{gathered}[/tex]

**Final Answer**

Hence ,

a) area of the floor is 104 square feet.

b) area of the floor is 14976 square inches.

c) Number of tiles required to cover the floor is 234 tiles

**Answer:**

a. Area of the floor = 104ft²

b. 14976 square inches

c. 234 tiles

**Step-by-step explanation:**

a. length = 13ft

breadth = 8ft

area = l×b

=13×8

=104ft²

b. 1ft = 144 square inches

104×144

14976 square inches

c. number of tiles = area of floor ÷ area of a tile

area of tile = l×b

= 8×8

= 64 square inches

= 14976 ÷ 64

=234 tiles

3. To break the school's record, the girls' time had to be faster than 12 2/5minutes. Did the girls break the record? If so, how much faster werethey? If not, how much slower were they?

### Answers

ANSWER

**The girls did not break the record**

**STEP-BY-STEP EXPLANATION:**

Given information

[tex]\begin{gathered} Barbara\text{ spent }3\frac{3}{10}\text{ minutes to complete the relay race} \\ \text{Donna spent 2}\frac{4}{5}\text{ minutes to complete the relay race} \\ \text{ Cindy spent x minutes to complete the relay race} \\ \text{ Nicole spent }2\frac{1}{10}\text{ minutes to complete the relay race} \end{gathered}[/tex]

From the question provided, the record set by the school is 12 2/5 minutes

The next step is to find the time spent by Cindy in the relay race

[tex]\begin{gathered} 3\frac{3}{10}\text{ + 2}\frac{4}{5}\text{ + x + 2}\frac{1}{10}\text{ = 12}\frac{2}{5} \\ \text{The next process is to convert the mixed fraction into an improper fraction} \\ \frac{33}{10}\text{ + }\frac{14}{5}\text{ + x + }\frac{21}{10}\text{ = }\frac{62}{5} \\ \text{let the co}mmon\text{ denominator be 10} \\ \frac{33\text{ + 14(2) + 10x + (21})}{10}\text{ = }\frac{62}{5} \\ \frac{33\text{ + 28 + 10x }+\text{ 21}}{10}\text{ = }\frac{62}{5} \\ \frac{82\text{ + 10x}}{10\text{ }}\text{ = }\frac{62}{5} \\ \text{Cross multiply} \\ 5(82\text{ + 10x) = 62 }\times10 \\ 410\text{ + 50x = 620} \\ \text{subtract 830 from both sides} \\ 410\text{ - 410+ 50x = 620 - 4}10 \\ 50x\text{ = 210} \\ \text{Divide both sides by 50} \\ \frac{50x}{50}\text{ = }\frac{210}{50} \\ x\text{ = }\frac{21}{5} \\ x\text{ = 4}\frac{1}{5}\text{ minutes} \end{gathered}[/tex]

The next step is to sum all the minutes spent in the relay race

[tex]\begin{gathered} \text{Total minutes =}\frac{33}{10}\text{ + }\frac{14}{5\text{ }}+\frac{21}{5}+\text{ }\frac{21}{10} \\ \text{Total minutes = }\frac{33\text{ + 28 + 42 + 21}}{10} \\ \text{Total minutes = }\frac{124}{10} \\ \text{Total minutes = 12}\frac{4}{10}\text{minutes} \end{gathered}[/tex]

From the above calculations, you will see that the girls ran a total of minutes of** 12 4/10 minutes**

Therefore, **the girls did not break the record**

a) what is KN+IKb)What is the coordinate of midpoint GO

### Answers

Given the figure of a number line

a) We will find the value of KN+IK

As shown in the figure:

[tex]\begin{gathered} K=0 \\ N=3 \\ I=-2 \\ So, \\ KN+IK=0\times3+(-2)\times0=0+0=0 \end{gathered}[/tex]

So, the answer will be KN+IK = 0

b) )What is the coordinate of midpoint GO

As shown:

[tex]G=-4,O=4[/tex]

The midpoint =

[tex]\frac{-4+4}{2}=\frac{0}{2}=0[/tex]

So, the answer will be the midpoint will be k = 0

What is the summation notation for the geometric series +2+8+32+128?1

### Answers

**Answer**

**Explanation**

For the geometric series given, we can see that the first term is

square root of 0.0625

### Answers

hello

the square root of 0.0625

[tex]\sqrt[]{0.0625}=0.25[/tex]

the square root of 0.0625 is 0.25

this is square root symbol

square of a particular number is always that number multiplied by itself

for example, the square of 2 = 2 * 2 = 4

square of 3 = 3 * 3 = 9

square of 10 = 10 * 10 = 100

now to find the reverse of this, we normally use a calculator but there's a method to do this without a calculator.

step 1

separate the number into two

step 2

now we need to find an integer whose square is less or equal to the left hand side of the divided number i.e (00)

in this case the number is 0

Note: i've already explained how to find the squares of numbers for you

now

An adventure company wants to run a zipline from the top of one building that's 130 ft tall to to the top of another building that is 30 ft tall. The two buildings are 72 ft apart. Estimate the length (in feet) of the zip line. Round your answer to the nearest tenth.

### Answers

[tex]We\text{ will graph the situation}[/tex][tex]\begin{gathered} By\text{ pythagoras theorem} \\ \\ 72^2+100^2=z^2 \\ 5184+10000=z^2 \\ 15184=z^2 \\ z=\sqrt[]{15184} \\ z=123.2233 \\ \\ \text{Thus the length of the zip line is about 123.2ft} \end{gathered}[/tex]

(1 point)(You will need a calculator.) Suppose you deposit $134,000 into an account paying 7.7% simple annual interest, and that you won'ttouch the account until it-holds $1,000,000. How long must you wait?Answer:years. (Round your answer to the nearest year.)

### Answers

The simple interest formula is as follows:

[tex]I=P\cdot r\cdot t[/tex]

Where P is the principal amount (the initial amount), r is the annual rate and t is the time in years.

The final amount is the initial amount plus the interest, so:

[tex]\begin{gathered} V=P+I \\ V=P+P\cdot r\cdot t \\ V=P(1+rt) \end{gathered}[/tex]

We have the princiapl value $134,000, the final value $1,000,000 and the rate of 7.7%, so:

[tex]\begin{gathered} V=1000000 \\ P=134000 \\ r=7.7\%=0.077 \end{gathered}[/tex]

So, we can solve for *t* and input the values:

[tex]\begin{gathered} V=P(1+rt) \\ \frac{V}{P_{}}=1+rt \\ rt=\frac{V}{P}-1 \\ t=\frac{\frac{V}{P}-1}{r} \end{gathered}[/tex][tex]\begin{gathered} t=\frac{\frac{1000000}{134000}-1}{0.077} \\ t=\frac{7.4626\ldots-1}{0.077} \\ t=\frac{6.4626\ldots}{0.077} \\ t=83.93\ldots\approx84 \end{gathered}[/tex]

So, you must wait **approximately 84 years**.

A salesman earns a commission of $350 for selling $2500 in merchandise. Find the commission rate. Write your answer as a percentage.

### Answers

14%

**Explanation**

we can solve this by using a rule of three

let x represents the rate commision( in percentage)

so

if

[tex]2500\rightarrow100\text{ \%}[/tex]

then

[tex]350\rightarrow x[/tex]

now, set the proportion and solve for x

[tex]\begin{gathered} \frac{2500}{100}=\frac{350}{x} \\ \text{cross multiply} \\ 2500x=350\cdot100 \\ 2500x=35000 \\ \text{divide both sides by 2500} \\ \frac{2500x}{2500}=\frac{35000}{2500} \\ x=14 \end{gathered}[/tex]

it means, the rate is 14%

I hope this helps you

**Answer:**

13%

**Step-by-step explanation:**

A positive B negative C zeroD the parabola opens upE the parabola opens up F the parabola opens down

### Answers

Solution

Step 1:

The general parabolic function is given as:

[tex]\text{y = ax}^2+bx+c[/tex]

If a is positive, the parabola is upward.

If a is negative, the parabola is downward.

Step 2

The c-value is where the graph intersects the y-axis. In this graph, the c-value is -3. The graph of a parabola that opens up looks like this. The c-value is where the graph intersects the y-axis.

Step 3:

Finally, the c-value can also be called the y-intercept of the parabola. Algebraically, this is where the x-value is zero and graphically, this is where the graph intersects the y-axis.

Final answer

**The c value of the function represented in the graph is -3 because the parabola opens up.**

What is the probability that a card drawn randomly from a standard deck of 52 cards is a king? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

### Answers

SOLUTION:

Case: Probability (52 cards)

A standard 52-card deck comprises 13 ranks in each of the four French suits: clubs (♣), diamonds (♦), hearts (♥) and spades (♠). Each suit includes three court cards (face cards), King, Queen and Jack, with reversible (double-headed) images. Each suit also includes ten numeral cards or pip cards, from one to ten. The card with one pip is known as an Ace. Each pip card displays the number of pips (symbols of the suit) corresponding to its number, as well as the appropriate numeral (except "A" for the Ace) in at least two corners.

Given: A standard deck of 52 cards

Required: Probability of selecting a king

Method:

Step 1: The probability formula:

[tex]Pr(event\text{ A\rparen}=\frac{n(A)}{Total}[/tex]

Step 2: The chances of obtaining event A (selecting King) is 4 out of 52.

[tex]\begin{gathered} Pr(King)=\frac{4}{52} \\ Pr(King)=\frac{1}{13}OR\text{ 0.019231} \end{gathered}[/tex]

Final answer:

The probability of selecting a 'King' is:

[tex]\frac{1}{13}\text{ }or\text{ 0.019231}[/tex]

( algebra )solve for the equation below for the indicated variable f=ma (solve the equation for m )

### Answers

Given the equation,

[tex]f=ma[/tex][tex]\begin{gathered} f=m\times a \\ \frac{f}{a}=m \end{gathered}[/tex]

Therefore,

[tex]m=\frac{f}{a}[/tex]

Find the exact length, in units, of the hypotenuse of the right triangle, shown below. Write your answer in simplified radical form.

### Answers

Right Triangles

A right triangle is identified because it has an angle of 90° (marked as a little square).

In a right triangle, there is a larger side called the hypotenuse, and two shorter sides called the legs.

Each one of the acute angles in a right triangle has an adjacent leg and an opposite leg. For example, the angle of 45° given in the figure has 15 as the adjacent leg. The hypotenuse is below it, and we'll call it H.

There is a trigonometric ratio called the cosine that relates the adjacent leg of an angle with the hypotenuse as follows:

[tex]\displaystyle\cos \theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}[/tex]

Applying to the given triangle:

[tex]\cos 45^o=\frac{15}{H}[/tex]

Solving for H:

[tex]H=\frac{15}{\cos45^o}=\frac{15}{\frac{\sqrt[]{2}}{2}}=15\cdot\frac{2}{\sqrt[]{2}}\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}}=15\sqrt[]{2}[/tex]

Hypotenuse length:

[tex]15\sqrt[]{2}[/tex]