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### Ratios in Right Triangles

- Trigonometry: The study of involving triangle measurement
- sine (sin) = opposite/hypotenuse
- cosine (cos) = adjacent/hypotenuse
- tangent (tan) = opposite/adjacent
- SOHCAHTOA

### Ratios in Right Triangles

^{ − 1}

^{o}.

^{o}

^{o}.

- m∠ A = tan
^{ − 1}0.8

^{o}.

Write sin M, cos P, and tan P.

- sin M = [NP/MP]
- cos P = [NP/MP]

AB = 2, BC = 4, AC = 5, find cos A.

m∠ M = 60

^{o}, find tan P.

- m∠ P = 30
^{o}

^{o}= 0.577.

In a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle.

cos A = 0.6, find m∠ A.

^{ − 1}0.6 = 53.1

^{o}.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Ratios in Right Triangles

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Trigonometric Ratios 0:08
- Definition of Trigonometry
- Sine (sin), Cosine (cos), & Tangent (tan)
- Trigonometric Ratios 3:04
- Trig Functions
- Inverse Trig Functions
- SOHCAHTOA 8:16
- sin x
- cos x
- tan x
- Example: SOHCAHTOA & Triangle
- Extra Example 1: Find the Value of Each Ratio or Angle Measure 14:36
- Extra Example 2: Find Sin, Cos, and Tan 18:51
- Extra Example 3: Find the Value of x Using SOHCAHTOA 22:55
- Extra Example 4: Trigonometric Ratios in Right Triangles 32:13

### Geometry Online Course

### Transcription: Ratios in Right Triangles

*Welcome back to Educator.com.*0000

*For this lesson, we are going to continue on right triangles.*0003

*We are going to go over ratios and right triangles.*0006

*Trigonometric ratios: first of all, trigonometry is the study involving triangle measurement.*0010

*Because we are going to go over trigonometric ratios, this all has to do with trigonometry.*0024

*Now, I know that you are probably thinking that this is geometry, not trigonometry.*0030

*But it involves a lot of trigonometry, because of the triangles.*0034

*And we are going over right triangles, so for this section, and for the next couple of sections, we are going to be using a lot of trigonometric ratios.*0041

*The most common trigonometric ratios are these three right here.*0051

*The first one is sine; it is pronounced "sign"; this is how you spell it out: S-I-N-E...but we always write it as "sin," but it is pronounced "sign."*0056

*The next one is cosine, but we shorten it as "cos"; but we call it "cosine."*0071

*The next one is tangent, but we only write "tan."*0082

*Now, one thing to remember about trigonometric functions: there are more than three, but these are the main ones that we are going to go over.*0089

*These are the most common; and for now, we are only going to use these three.*0100

*But these, we can only use with angle measures; it is very, very important to remember that we can only use*0106

*the sine of an angle measure, the cosine of an angle measure, and the tangent of an angle measure.*0114

*They never stand alone; they always have to go with an angle measure.*0121

*So, if I say "the sine of 90 degrees," that is one way that I would say it; I am looking for the sine of 90 degrees.*0124

*Or I can say "cosine of 45 degrees," "tangent of 60 degrees..."*0138

*Always remember that sine, cosine, and tangent must be with an angle measure; you can only find the sine of an angle measure.*0149

*These trigonometric functions don't stand alone; they are with angle measures, and only angle measures.*0159

*You can't find the sine of just any random number; it has to be the sine of angle measure; that number must be an angle measure.*0167

*So, if you have a scientific calculator, or if you have a calculator like this, you are going to need it.*0175

*And we are going to practice finding values on the calculator.*0186

*Here, trigonometric functions, again, are sine, cosine, and tangent.*0194

*If you look on your calculator, you should have buttons that say "sin," "cos," and "tan."*0200

*Now, these right here, inverse trigonometric functions: if you look at the same three buttons,*0209

*then above it, it should say "sin ^{-1}"; above the "cos," it says "cos^{-1}"; and above "tan," it says "tan^{-1}."*0217

*Those are very, very important; we are going to practice using those key functions here.*0230

*We have sine, cosine, and tangent; and then above it, the second key is inverse sine, cosine, and tangent.*0240

*Now, when do we use each of these?--again, for these, we are going to use angle measures.*0250

*So, if I wanted to find the sine of 60 degrees, then I would punch in "sin(60)"; and you are going to get a number.*0256

*And that number says 0.866; so the sine of this angle measure equals this.*0276

*That is when you use trigonometric functions: the sine, cosine, or tangent (depending on what you have to find)*0291

*of the angle measure equals...and that is what your calculator is going to give you, the answer.*0298

*Now, when do you use these? Well, if you punch in "sin" and any number right after sine--*0302

*if you punch in sin, cos, or tan, and you punch in a number after sin, cos, or tan--*0311

*the calculator is going to think--it is going to assume--that that number that you punched in is an angle measure,*0318

*because again, you can only find those functions of angle measures.*0324

*If I have, let's say, the sine of x equals 0.866--so I have the answer, and I am missing the angle measure--*0333

*that is what I am missing, so that is where it has to go, here, because angle measure always has to go there--*0345

*always, always, always, it is the sin of an angle measure--because I don't have the angle measure*0351

*(I only have what it equals), I can't plug in this number here, because this is not the angle measure.*0357

*So, if I punch in the sine of 0.866, the calculator is going to think that .866 is the angle measure, which it is not.*0365

*This is the answer; I am looking for the angle measure.*0378

*So, depending on what you have, you would have to use different things.*0382

*Now again, if you punch in sin(60), the calculator will know that 60 is the angle measure,*0385

*and therefore, the calculator is going to give you the answer to that, the sine of that angle measure.*0393

*If you want the calculator to give you the angle measure (you are not giving the calculator the angle measure--*0400

*you want the calculator to give you the angle measure), then that is when you have to use the inverse sine.*0407

*You are telling the calculator, "Well, I have this--I have the answer 0.866;*0414

*I don't have the angle measure to give you; given the answer, I want the angle measure."*0421

*Then, you would, second, press the sine (on your screen, is should say that it is the inverse sine); and then you punch in 0.866.*0427

*By doing that, the calculator now knows that that number that you punched in is not the angle measure.*0442

*Then, the answer is 59.997, which is 60 degrees; so let's just write 60.*0448

*Now, that is the angle measure; it is really important to remember which one you have to use.*0468

*Whatever number you punch in after sine, cosine, or tangent needs to be the angle measure, and the calculator will assume that.*0474

*So, if it is not the angle measure (you want the calculator to give you the angle measure), then you have to do inverse sine,*0482

*so that the calculator will know that that number is not the angle measure (it needs to give you the angle measure).*0488

*Let's just do a few of those...oh, before that, we are going to go over this right here, "Soh-cah-toa."*0495

*Soh-cah-toa is just an easy way for you to remember three formulas.*0507

*Here we have "soh"; "cah" is another formula, and so is "toa."*0518

*Now, I know that it sounds funny; but just say it to yourself a few times, so that you get used to this word, "Soh-cah-toa."*0530

*And that is going to help you remember three of the formulas, which are also known as the three ratios.*0537

*Each of these stands for something: S is sine (we are going to write down this formula here).*0549

*Sine of...again, it has to be an angle measure, so let's write...x, equals...o is for opposite, the side opposite;*0558

*the opposite side, over...h is for hypotenuse; so all of this "soh" is this right here: "Sine of x equals opposite over the hypotenuse."*0570

*That is the "Soh," and that is the ratio for sine.*0592

*Now, for cosine, it is right here: "cah" is going to be "Cosine of x is equal to"...the a stands for adjacent side, over the hypotenuse for the h.*0600

*Cosine of x equals the adjacent side over the hypotenuse side.*0627

*And the last one, "toa" is for tangent: Tangent of x equals...o is for opposite side, over...a is for adjacent side.*0634

*Then, these three right here are the actual trigonometric functions; and then the rest,*0659

*the "oh," "ah," "oa," are all for sides: o is for opposite side; a is for adjacent; and h is for hypotenuse.*0666

*So then, again, "soh" is "Sine of x equals opposite over hypotenuse."*0678

*This one is "cosine of x equals adjacent over hypotenuse," and then, this one right here is "tangent of x equals opposite over adjacent."*0687

*This will help you remember these three formulas; and that is what this was for.*0697

*Sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; and then, tangent of x equals opposite over adjacent.*0703

*Again, x is going to be the angle measure--only an angle measure can go there.*0731

*So, let's say C: we are going to find sine of C.*0735

*That is C, so now we are talking about from this angle's point of view.*0748

*From this angle's point of view, what is the side opposite?--because we are looking for the side opposite.*0752

*The opposite side would be side AB; and "over hypotenuse"--what is the hypotenuse? BC, so it is over BC.*0762

*So again, the sine of C (now, it doesn't always have to be the sine of C; they will either name the angle,*0777

*or they will tell you from what angle's point of view), for this one, we are going to do angle C's point of view: what is the side opposite?*0785

*It is that right there, over the hypotenuse, which is BC.*0796

*And then, for the cosine of C, again, from angle C's point of view, the adjacent side is the side that is next to it--*0802

*not that one that is opposite, but it is the one that is next to it; it is the other leg, the AC.*0813

*The side adjacent to angle C is AC, over the hypotenuse, which is BC.*0821

*And then, the tangent of C is going to be opposite (which is AB), over the adjacent, which is AC.*0830

*So again, the sine of x equals opposite over hypotenuse; cosine of x equals adjacent over hypotenuse; tangent of x equals opposite over adjacent.*0845

*This is really, really important to know; again, just say this a few times to yourself: "Soh-cah-toa."*0860

*And that will definitely help you, because you do need to know these three ratios.*0867

*Now, let's do a few practices on the calculator, finding trigonometric functions.*0879

*We are going to find the value of each ratio, or the measure of each angle.*0888

*The first one, the sine of 15 (now, this is an angle measure, so let me just do that--these are angle measures), equals...*0893

*and that is what we are looking for: so you go on your calculator, and just punch in "sin(15)".*0905

*And then, because you are writing "15" right after you punch "sin," the calculator knows that 15 is the angle measure.*0916

*Then, the answer becomes .2588.*0923

*And then, the next one: it will be the tangent of 72, and that becomes 3.0777.*0936

*I am just rounding it to four decimal places: write that over here...3.0777.*0953

*Now, here, we don't have the angle measure; the angle measure is e--that is what we want to find.*0965

*We want the angle measure; so when we punch it in, we want the calculator to give us the angle measure.*0974

*But if you punch in "tan(0.9201)," the calculator will think that this is the angle measure, which is not true.*0979

*This is not the angle measure; e is the angle measure.*0987

*So, you tell the calculator, "I am going to give you the answer, and I want you to give me the angle measure."*0990

*Inverse tangent, remember, is 2 ^{nd} and then tan: .9201...equals...*1001

*and then, the calculator knows that you want the angle measure, and that is 42.6 degrees.*1011

*The same thing here: we don't have the angle measure; we want the angle measure; that is what we are looking for.*1027

*When you punch it in, you can't punch in cosine of this number, because then the calculator...*1033

*now, let's just try it: just try clearing your screen, and then just punch in "cos(.2821)".*1039

*Now, it is going to give you .999987 and so on; that rounds to 1.*1057

*Now, this number is not the angle measure, because the calculator, because you punched in cosine of this number,*1067

*would assume that this is the angle measure; and so, it is going to give you the answer if this were to be the angle measure.*1078

*But what we have to do is tell the calculator that that number is not the angle measure.*1088

*Convert it to inverse cosine, and then the calculator will give you the angle measure: 73.6 degrees.*1101

*Just be very, very careful with that: if you have a number here, that would be the angle measure.*1111

*If you have a variable there, then you are doing the opposite: you are looking for the angle measure.*1118

*Make sure you punch in inverse trigonometric functions.*1126

*The next one: Find sine of a, cosine of a, and tangent of a for each.*1133

*Here is where we are going to be using Soh-cah-toa.*1140

*This one right here is going to be "sin(a)," the angle measure, "equals"...don't forget: if you don't have the angle measure,*1153

*make sure that you write a variable right there; you can't leave it as "sin() ="; there has to be something there.*1163

*The sine of a equals opposite over hypotenuse; cosine of a is equal to adjacent over hypotenuse;*1169

*and then, tangent of a is going to be opposite over adjacent.*1184

*The first one: to find this, we are going to have to use the sine one: sine and sine.*1195

*Then, sin(a) here is going to be opposite (from a's point of view, what is the side opposite?*1201

*It is this, so what is the measure of that?)--it is 5, over the hypotenuse, which is 13.*1218

*Now, you are just going to leave it like that, because it is a fraction, and you can leave fractions: sin(a) = 5/13.*1228

*The next one, cosine of a: again, from A's point of view, it is going to be adjacent; there is the adjacent: 12,*1235

*the one next to the angle, over the hypotenuse, which is 13.*1251

*And then, tangent of a is opposite over adjacent: from this angle's point of view, opposite is 5; adjacent is 12.*1257

*Now, the next one: let's look for a right here, again, from A's point of view.*1276

*So then, sine of a is opposite (which is 3), over hypotenuse (which is 5).*1281

*Cosine of a equals adjacent (which is 4), over the hypotenuse (which is 5).*1296

*And then, tangent of a is opposite (3) over the adjacent (which is 4).*1306

*That is all they wanted you to find--just those things.*1318

*Now, if they wanted you to find the actual angle measure, that is different;*1320

*then you would have to use your calculator and do the inverse sine,*1326

*because again, this is not the angle measure, so you can't punch in the sine of this number.*1330

*So, you are going to have to do inverse sine, so that you find the angle measure.*1335

*But then, here, for this problem, you don't have to find the angle measure, because they just want you to find sin(a), cos(a), tan(a)--that is it.*1340

*We found sin(a), cos(a), and tan(a) for each of these triangles; that is it.*1347

*In the next example, it is going to ask you for the actual angle measure.*1354

*So, you are going to have to use these trigonometric ratios to actually find missing sides and angles.*1359

*You are actually solving for something there; in this, they just wanted you to actually just write down the ratio; that is it.*1366

*Next, find the value of x.*1375

*Like the previous example, we are not going to use...*1383

*well, in the previous example, we used all three trigonometric ratios, because they wanted us to find sin(a), cos(a), and tan(a).*1387

*For this one, we don't have to use all three; we just have to use whatever we need in order to find x.*1395

*We have to first look for an angle's point of view.*1405

*So, look for an angle: this is the angle that is given, so we are going to use this angle right here.*1409

*And then, what sides of the triangle are we working with?*1416

*Are we working with the opposite? No, we don't have the measure of the opposite one.*1421

*We have the adjacent, and we have the hypotenuse.*1428

*Now, if I were to write out Soh-cah-toa again, just so it is easier to see,*1432

*the three ratios are going to be sin(x) = opposite/hypotenuse; the next one, cos(x), equals adjacent/hypotenuse;*1439

*and then, the last one is tan(x), equal to opposite/adjacent.*1459

*Which one of these three would we use?*1473

*We don't have the opposite; we only have the adjacent, which is this side right here*1480

*(because that is what we are looking for, so we have to include that one) and the hypotenuse.*1486

*Which one uses, from this angle's point of view, the adjacent and the hypotenuse?*1492

*Opposite over hypotenuse...no; adjacent over hypotenuse: so then, we are going to have to use this one right here; we are going to have to use cosine.*1498

*Now, do we have the angle measure--do we have x?*1509

*Yes, we do, so now we are going to just start plugging in these numbers for the angle measure, the side adjacent, and the hypotenuse.*1513

*Cosine of 55 equals adjacent (what is the one adjacent? That is x), over (what is the hypotenuse?) 12.*1522

*Now, we are going to go to our calculator, and we are going to find x.*1538

*Cosine of 55: now, 55 is the angle measure, so I can just punch in cos(55); cosine of 55 is .5736.*1546

*That equals...all of this is equal to this; that equals x/12.*1568

*Now, how do I solve for x here?--I am going to have to multiply the 12: multiply 12 on that side, and then multiply 12 to this side.*1573

*Then, x becomes...I just have to multiply: instead of clearing it, I can just leave that number, and then just multiply it to 12.*1583

*And I get 6.8829: so, this right here, this length, is 6.8829.*1595

*Again, to go over what we just did: cos(55)...now, why did we use cosine, and not sine and not tangent?*1617

*It is because we have to look at what we have.*1625

*And again, we don't have to use all three of them; we just have to use the one that we need,*1629

*unless it is like the previous problem, where it asks for all three; this one is not--it is just asking to find the missing values.*1634

*So, you have to look for cosine; you have to use cosine, because from this angle's point of view, you only have the adjacent and the hypotenuse.*1641

*So, you are going to use cos(x) = adjacent/hypotenuse; cos(55) =...adjacent is x; hypotenuse is 12.*1656

*You punch this into your calculator; you get this; then you multiply 12 to both sides, and you get the answer.*1666

*Now, I can find this third angle measure by subtracting it by 180, or I can just do 90 minus this number, and then I get this number.*1675

*I could do that, and then, if you want, you can use this angle's point of view.*1687

*This right here is actually going to be 35 degrees; and then, instead of using 55, you can use this one.*1692

*If you decide to use that one, then it is a different perspective, a different point of view.*1705

*So then, what would you have to use?*1710

*We have opposite (this is opposite), and then the hypotenuse; if you are going to use this angle, then you would have to use the sine,*1713

*because sin(35) is opposite over the hypotenuse; so you have options.*1722

*You don't have to do both; you just have to use one of them.*1731

*Now, because this is the angle that is given, it would just be easier, instead of doing more work, to look for this angle, and then go on from there.*1733

*But either way, that is an option for you, if you would like to just use that angle instead.*1743

*Let's do the next one: here we have...*1751

*And we are going to have to use this one; it is not like this one, where we have options,*1756

*where we can use this angle or this angle; for this one, we have to use this angle, because that is the angle that we are looking for.*1760

*And then, plus, we can't subtract it from 180 (because we don't know what this angle is) to find that angle.*1765

*We have to use this angle; from this angle's point of view, you have to work with the side opposite and the hypotenuse.*1771

*So, you have opposite, and you have hypotenuse; what are you going to use?*1785

*With opposite and hypotenuse, you are going to have to use sine, because that uses the side opposite and the hypotenuse.*1790

*The other ones use adjacent and the hypotenuse (no, we don't have the adjacent);*1801

*this one uses opposite, which we have, but no adjacent; so we have to use the sine.*1806

*Then, sine...and then, what would go as the angle measure?*1814

*We don't have the angle measure; that is what we are looking for.*1820

*So, we are going to put x right there; that equals...the side opposite is 21, over...what is the hypotenuse?...25.*1822

*Then, from here, I am going to just go to my calculator.*1835

*And again, don't punch in sine of this number, because then your calculator would think that this is the angle measure.*1842

*But it is not; so you are going to do inverse sine; that is 2 ^{nd} and sine; you should see that sin^{-1}.*1851

*And then, 21 divided by 25...and then, you should get...*1865

*That automatically just gives you x, because you already used the inverse sine.*1875

*57.1 degrees: that is this angle measure; 57.1 degrees--that is x.*1881

*That is it; just remember that you have to look for an angle.*1897

*That way, you know from which angle's perspective you have to look to see what you have.*1905

*And then, from there, based on what you do have, what sides are given, and what you are looking for,*1912

*you are going to have to pick one of these: sine of x, cosine of x, or tangent of x.*1918

*And then, using one of those ratios, you are just going to plug in the numbers, and then solve for whatever missing value you need to solve for.*1924

*Let's do the last example: this one is a little bit tricky, but it is not difficult at all.*1932

*Just remember that, when you are given points like this, and you have to find the measure of angle B,*1944

*now, when you are given this problem, you don't know if you have to use trigonometric functions yet.*1953

*So, let's just plug in the points first and see what we are dealing with.*1963

*The first one, A, is (-1,-5), right there; and then, B is -6 (1, 2, 3, 4, 5, 6) and -5; and then, C is -1, 1, 2, right there.*1972

*So, here is the triangle; I know that this is a right angle.*2001

*I know that because here, this point was (-1,-5); this is -5; and this point right here was (-6,-5).*2013

*So then, these two points are on the same y-value right there, so then it makes a horizontal line.*2028

*And then here, since this is (-1,-5), and this is (-1,2), they share the same x-coordinate, so this is a vertical line.*2039

*A horizontal and a vertical--they are perpendicular.*2050

*Now, what was this point? This point was A; this was B; and then, this right here is C.*2055

*And we want to find the measure of angle B; so then, this is the angle that we have to find.*2067

*Now, this is our variable; and then, let's look for side length.*2076

*Because here, it is perfectly horizontal or perfectly vertical, we wouldn't have to use the distance formula--we can just count.*2087

*Now, when it comes to BC, I am going to have to use the distance formula if I want to find BC.*2097

*Why?--because it is going diagonally, and you can't start counting to see the distance when it is going diagonally.*2104

*When it is going horizontally or vertically, then you can; you can just count the units.*2113

*So here, this is (-1,-5), and this one is (-6,-5); from -6 all the way to -5, this is 5 units.*2120

*If you have a bigger graph, you can start counting: it would just be from here, 1, 2, 3, 4, 5; so this is 5.*2136

*Then, this right here, the vertical--we can count that also, so here is -5: 1, 2, 3, 4, 5, 6, 7; this is 7.*2146

*AB is 5; AC is 7; we don't know BC.*2160

*Now, if you want, you can go ahead and solve for BC by using the distance formula.*2165

*The distance formula is x _{2}, the second x, minus the first x, squared, plus (y_{2} - y_{1})^{2}.*2170

*That is the distance formula; so you can just, using the coordinates for B and using the coordinates for C,*2183

*just plug it into this formula to find the distance of B to C.*2188

*But I don't think I will need it; now, let me re-draw this triangle...here is A, B, C.*2194

*This is what I am looking for, here; this is 5, and this is 7.*2209

*Now, if I am looking for this angle measure, what do I have to work with?*2216

*I have the side opposite, which is 7, and I have the side adjacent.*2220

*So, I know that I am going to use...Soh-cah-toa: from here, which one uses opposite and adjacent?*2231

*That would be this one right here, so we are going to use tangent.*2246

*That means that tangent of...I am just going to use b for the variable, because that is the angle that they want us to find...*2250

*tangent of b equals opposite (what is the side opposite? It is 7), and...what is the side adjacent? 5.*2260

*So then, here is my equation: tan(b) = 7/5.*2274

*Then, you just go straight to your calculator.*2279

*Now, remember: don't forget; don't push tan(7/5), because that is not the angle measure; b would be the angle measure.*2284

*Push 2 ^{nd}, tan, to get the inverse tangent, 7/5.*2290

*I just close the parentheses; then, the calculator knows that I am giving them this answer, and that I want the angle measure.*2299

*The angle measure that it gave me was 54.46 degrees.*2313

*And that is it; now, when do we use Soh-cah-toa--when do we use trigonometric functions?*2321

*First of all, you must have a right triangle; they must be right triangles.*2335

*And #2: when you are dealing with angles and sides together (you are given an angle measure,*2344

*but you are looking for a side, using that angle measure; or given sides, you have to look for an angle measure)--*2354

*anything that uses a combination of angles and sides of a right triangle is when you are going to use Soh-cah-toa.*2360

*What if they give you two sides, and they just want you to find the other side, the missing side, the third side?*2368

*Well, we don't have to use Soh-cah-toa, because no angles are involved.*2377

*It is just only the sides; so let's say they gave you that this is 5 and this is 7, and they wanted you to find the missing side.*2381

*Now, do we have to use Soh-cah-toa here--do we have to use trigonometric functions here?*2397

*No, because no angles are involved here; then what can we use?*2401

*We can use the Pythagorean theorem, remember, because 5 ^{2}, a^{2}, plus b^{2}, equals c^{2}.*2409

*Again, you are only using Soh-cah-toa when you are given sides, and then they want you to find the angle measure;*2418

*or given the angle measure, to find the missing side; and so on.*2426

*We are going to continue trigonometric functions next lesson.*2431

*For now, thank you for watching Educator.com.*2435

0 answers

Post by Kevin Yuan on February 16, 2014

I always forget which one to use even though there is the sohcahtoa. Any advice?

0 answers

Post by Dania Aljilani on November 4, 2013

Thank you soooo much! My math teacher spent three periods trying to explain this and I understood it in one lecture. You are a great teacher!

2 answers

Last reply by: Denise Bermudez

Wed Mar 11, 2015 5:50 PM

Post by Matthew Johnston on August 17, 2013

I am putting everything you are and I am getting wrong answers everytime??