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$D$ is a point on $A C$ of the triangle with vertices $A(2,$,3), $B(1,-3), C(-4,-7)$ and $B D$ divides $A B C$ into two triangles of equal area. The equation of the...

Question

$D$ is a point on $A C$ of the triangle with vertices $A(2,$,3), $B(1,-3), C(-4,-7)$ and $B D$ divides $A B C$ into two triangles of equal area. The equation of the line drawn through $B$ at right angles to $B D$ is(A) $y-2 x+5=0$(B) $2 y-x+5=0$(C) $y+2 x-5=0$(D) $2 y+x-5=0$

$D$ is a point on $A C$ of the triangle with vertices $A(2,$, 3), $B(1,-3), C(-4,-7)$ and $B D$ divides $A B C$ into two triangles of equal area. The equation of the line drawn through $B$ at right angles to $B D$ is (A) $y-2 x+5=0$ (B) $2 y-x+5=0$ (C) $y+2 x-5=0$ (D) $2 y+x-5=0$



Answers

In Exercises 59-62, the points represent the vertices of a triangle. (a) Draw triangle $ABC$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $AC$, and (c) find the area of the triangle.

$A = (0, 0), \quad B = (4, 5), \quad C = (5, -2)$

Okay, so we're act if I ng for actually have to draw the triangle. ABC, this is our team got out. We have negative 1/2 made. It would have been approximately actually down here somewhere. A the two somebody and he finds, or the two witches. 2.5. Calm down. We have trying. Oh, that looks like it. Now for part B were asked to find out food from over 60. You can go of the child with society, eh? So first, we're gonna have to find the line of between the points A and C. We're going to find support party. I find it. Look for M is equal to why? To minus one. Your old Linus Quint. A point from why are part of being witches? That's why I have over. I want to minus 1/2 that minus minus one house. That's plus 1/2 that usually 1/2 over six over to give me this one over. Okay, now that we have a slope like that into one of our point A to point B and then wouldn't write that in points for that's what I wanted. Why one that people don't people too I'm looking forward. Time minus. Explaining what? Your Bible. What I get? Why people like when it's five over 12. Well, let's put this in point for in our general form. So I'm gonna move, like to decide. So we're gonna have one. Over. What? Why? Minus five over 12. You know, now that we have in general, for we can find our altitude, which is the distance between this line and the point B, which is to calm a three. You know, that we said he was equal to accommodate This is gonna be excellent. Why one And then our dental form of the line is one over minus wine when a lie will fall if he could go. So a is equal to one other thing Be equal to want made of one. He is equal say, in the asylum. But we have our height. It is equal to a distance with his actual. Are you okay? One over six times. One would but B which is made of one times three. What I want. Did you see all over a word? One over word. Be quite like this one. That way we have one of those I do that minus okay. 512 wanted by squarely, uh, one squared that one. What does that give me? That approximately three point field for one. All right. No. That we find on our height are you for part C West to find the area of the triangle. Well, if you look back here, we found our altitude from a two point here that our height here. So we didn't find the base, which is a feeling of A and C. So what we just have to find is based, but the secrets of the distance between A and C that neither one but there are great, but, uh, finally, with two that's made of one over. Give me a bit. Actually, I'm inside it. The square root of one over four. Um, that gives me two squared with this one. So one on earth, born more. Hey, back to grow out of that, we get to a point. Okay, So we have our area and people to 1/2 our base point terms are height, which is three point. So for once, give me wait. Get area 3.13 fine.

And this question were asked to find the distance between A and B of the length of line segment A B, the length of line segment BC, and in the length of line segment A C also asked to determine if these points or Colin ear So I'm actually gonna start by plotting all three points a total of five negative five here that'll be point a point B is a 05 So that will be here and then point c is that point to one in my eyeball? That and look like to see that it looks like they're pretty Kalin ear. But let's just double check here. If we go from B to C, we go down. 1234 over, too. And then if we go down 1234 over. Two more Down 12 So it seems like we should be able to go down to over one down to over one, down to over one. We hit Ah, here, right. We say they're t connect B to C to take me to see we've got ah, slope here of down to over one. And so if we continue that to go from sea to aid down to over one, down to over one, down to over when we actually do hit see. So the slopes of those two components are those two pieces are the same. So it would appear that the points air Colin ear. So yes, they are Colin Ear. And we would say that see Point C is between A and B. All right, so that answers the second part of the question. Let's take a look at the first part of the question. The first part of the question asks us to find three distances. So we know that in general, are distance formula eyes equal to says that the distance between two points is equal to the square root of X two minus X one squared or the difference between the X coordinate squared and why two minus y one squared the difference between the Y coordinates square. So we're gonna take point a beast. Let's start with that points a to B and we're gonna say the distance there would be the square root of, ah, take the X coordinates the difference between the X coordinates, which would be five minus zero square that let me take the difference between the corresponding why Coordinates as well. So negative. Five minus five squared. So, essentially, I'm calling this X one. Why one on f? Rather, I called this X to y two on duh for point B. I called this X one y one Doesn't really matter which one's X one and why one a CZ Long as you're consistent in the directions off because I took the A coordinate minus toe be coordinate. I want to do that in both cases. So let's see how simplify here we're gonna have the square root of five minus zero is just five squared. Plus, here, we're gonna get negative. 10. But we want to remember that it's gonna be negative. 10 square. So we're gonna take negative 10 times. Negative. 10. This is a square root 25 plus 100. Or we can say that a B is the square root of 125. That's the distance there of a distance between B and C distance between B and C. We're gonna take again this time. We'll call this X two, and why, too? So we'll say, Let's see, what's the square root of Let's take the C coordinate, which is to minus the be coordinate and then the sea Coordinate to see why coordinate from point C and minus the y coordinate from point B and we'll square that. All right, so we have two minus zero, which is to swear one minus five that I would get me a negative four going to keep that in parentheses, because when I square it, I want to square the entire value. So I want to square the four and the negative. So two squared and the negative four times native for B positive 16. And so B C is the square root of 20. And there we have a B and B C. Last thing we need then is Ah, a c. So let's come over here. We'll do a C here and for a C. I'm gonna Let's see. Let's change this. We'll use this as X one. We'll make the A coordinates this time. The X one in the lie one and all right, so we're gonna take the square root for a C the square root of the X coordinate from see, which is to minus the X coordinate for A, which was five squared the Y coordinate from See, which is one minus a negative. Five squared. All right, so you've got to minus five. That'll be negative. Three squared, and then we have one minus and native, so it's gonna be plus five. So six square to hear. So be the square root of native three times a day. Three is 96 times six is 36. So here we've got the square root of 45 and that would be the value for a C.

According to Green question, the vortices off triangle is given as a joke. Me is for common ministry and sees one Goma minus five. No, we have to find the length on angle. So here we will say, this is a B and C. Now we can say that using distance formula that you can like that Rude under Xel minus X two holds good les Y one minus y to hold square. So according to this formula, we can night, Here s is a cool dude under four minus one whole squared plus minus three minus minus plus five full square. So this is nine plus four, which is Ruth Gordon. And so value of Ruth Gordon is is equal to 3.6. Now we can say that me is it called the road under zero minus one squared plus judo plus five. So this is one plus 25 which is Route 26. So we can say that Busy called 5.1 now, similarly cuz flavor now, according to the coach and all we can say that cause musical do is quite Lasi squared minus b squared. They were aided by to see So this is a squad is 13 C squared is 26 minus 25 divided by two rotunda 13 and rude under 26. Rude Under going to fight because this is five through 35 night. So this is the value of this is zero point 33 to it. So we can say that angle. He's a cool dude. 71 victory And using sign rule, we can calculate the other values as so. Since this is angle B, we can say that sign is equal to IHS Signed me divided by me So it signed me. So this would be 3.6 sign 71 degrees divided by B these 5.1 so well so we can say that angle is equal to 42 degrees. So now angle being a lady already know So angle see igniters one a. D minus 71 plus 42 days. So this is record to 67 degree So these are the value off angrily and will be an embassy

All right. Here we have a sketch of the triangle with the Given Vergis ese, and we're going to start by finding the length of each side using the distance formula. So from a to B, we would have the square root of the change in why would be one minus zero quantity squared? Plus, the change in X would be six minus negative, one quantity squared, so that simplifies to be the square root of one squared. So just one plus seven squared. So 49 that's the square root of 50. I'll just keep that exact. For now, the problem does not say to approximate, so A B is 50. We want to continue doing that same sort of thing to find the other two side lengths just going to need a little bit more room. I'm going to move on to do BC next. So for BC, we would have the square root of the change in the white coordinates would be three minus one. That's not what one looks like, three minus one quantity squared, and then the change in X would be zero minus six, and we need to square that as well. So we have the square root of two squared for, uh, plus negative six squared at 36 and that's gonna be the square root of 40. Now, we could simplify. In fact, if we go back to the square root of 50 we could simplify that and call that five square root two. Didn't really think of that before. That would be better. So square to 40 is the same as, um, let's see, 40 is four times 10. So that would be too square root. 10 for this is B C not just be, but BC Okay, moving right along. Now we need to find the length of side A c. So we're gonna go ahead and use the distance formula again. The change in why Between a and C, we have three minus zero and we square that and then we add the change in X squared. The change in X would be zero minus negative one. We square that So we have three squared nine plus one squared one swift square root 10. Okay, so we have the lengths of all three sides. That takes care of part, eh? Now, for part B, use the converse of the Pythagorean theorem to show that the triangle is a right triangle and identify the right angle. So our sides were Abie was five square. Two BC waas to square a 10 and a c waas square a 10. Let's go ahead and see if the shorter too squared added together equal the longest one squared. So let's see if it works in the Pythagorean theorem, so the shortest one would be. Square it 10 and the next one would be to square root. 10. So we have square 10 squared, plus two square root 10 squared and does that equal five square two squared. Let's check it out. Square 10 squared is 10 to square, 10 square to square the two you get. Four. Square the square, attend you get 10. So that's 40 and five square. Two squared square the 5 25 square square to two. So that's 50. So 10 plus 40 equals 50. Yes, that checks. So it is a right triangle. So what's the right angle? Well, the right angle is across from the longest side. The longest side was side A B, so that tells us the right angle must be angle. See Okay. Now, there is still one more part of this problem. Find the measures of the acute angles. So we're gonna find the measure of angle A and the measure Bengal beat round to the nearest degree. So once again, here's our sketch. Now, this is not the triangle that I pictured originally, but that's okay. As long as we have all the side lengths correct, we can get the right angle measures. So we have our five square root, too. And for BC, we have to square root 10. And for a C, we have square attempt. It doesn't matter that it's not drawn to scale or the correct orientation, since we have all the numbers we need. All right. So let's find the measure of angle, eh? So we could use sign or co sign or tangent. It doesn't matter. It'll work any way you choose. As long as you use the correct sides, I'm going to use the tangent of a That would be too square 10 the opposite over square root 10 the adjacent that does reduce to two. So the tangent of a is too. And then to find a you confined the inverse tangent of to using a calculator, and we're rounding this to the nearest whole number, so that works out to be 63 degrees. And then to approximate the value of the other acute angle, remember that the acute angles add 90 degrees. So a plus b measure Bengal, a plus measure bank will be equals 90 degrees. And now we know the measure of anklet is 63 approximately, so we can subtract that from 90 and that will give us the measure angle B, and so that's going to be approximately 27 degrees.


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