CA Geometry: Secants and Translations 76-80, secants and graph translations
CA Geometry: Secants and Translations
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- We're on 76.
- In the figure below, line AB is tangent to
- circle O at point A.
- Fair enough.
- Secant BD intersects circle O at points C and D.
- Secant just means that it intersects the
- point at two points.
- It's not tangent.
- A tangent intersects the circle at exactly one point.
- Secant BD intersects the circle at points C and D.
- Measure of arc AC is equal to 70 degrees, and the measure of
- arc CD is equal to 110 degrees.
- So what do they want us to figure out?
- What is the measure of ABC?
- So what is this angle?
- That's what they want us to figure out.
- They want us to figure out this right here.
- That angle.
- Let's see what we can do here.
- So there's a couple of ways to think about it.
- Well, if you look at it, this is 70, this is 110, right?
- So if you add them, this whole arc, going for all the way
- around, that's 180 degrees.
- So it's actually perfectly halfway around the circle.
- So actually from A to D is actually a
- diameter of the circle.
- How do I know that?
- Because it goes all the way around the circle.
- The arc length is 180 degrees.
- So we can draw-- let me draw a diameter line.
- I don't know if this thing is drawn to scale.
- A to D is actually a diameter of the circle.
- We know that, because that's 180 degrees, the combined arc
- length right there.
- And we know, that this line is tangent at point A.
- So that has to be a right angle.
- When you're tangent at point A, that means your tangent to
- the radius at that point.
- Fair enough.
- So now let's see what we can figure out.
- We know that this arc length is 70.
- Is there any way to figure out what this angle right here is?
- Because if we know this angle, we know this is 90, then we
- could figure out that angle.
- And this is something that you should just learn about
- inscribed angles in a circle.
- This is an inscribed angle, because the vertex touches on
- one of the sides of the circle.
- So an inscribed angle-- and this is just something good to
- memorize, and you could play around with it to get a little
- bit more intuition about it-- is equal to half of the arc
- length that it intersects.
- So this inscribed angle intersects an arc
- length of 70 degrees.
- So this is 35 degrees.
- And that's just something good to know.
- That this is going to be half of whatever
- this arc length is.
- Now we can figure out x, because x plus 35 plus 90 is
- equal to 180.
- You get x plus 35 is equal to 90. x is equal to 90 minus 35
- is 55, which is choice C.
- Now there's another way you can do this, and this is
- another interesting thing about lines that
- intersect in circles.
- It's that this angle measure, right here, if it's sitting
- outside of the circle-- I think there might be ones
- sitting inside the circle later on-- but if it's sitting
- outside of the circle, it's equal to 1/2 the difference of
- the arc measures that it intersects.
- So for example, it intersects this arc measure of 70, and it
- also has this arc measure right here.
- When it hits the circle and intersects at 70 degrees and
- when it exits the circle, it has this arc measure.
- This arc measure we already figured out.
- It's 180 degrees, because that's 180 degrees.
- So you know that this angle right here, you could say that
- x is equal to 1/2 the difference of these two arc
- measures, and we can do that because x is
- outside of the circle.
- If x was inside the circle, then x would be 1/2 the sum of
- the two measures.
- So what's the difference?
- It's 180 minus 70.
- So that's equal to 1/2 times 110, so that is equal to 55.
- So it's nice to see that math works correctly no matter how
- you do the problem.
- Problem 77: In the circle shown below, the measure of
- arc PR is 140 degrees.
- Sure enough.
- The measure of angle RPQ is 50 degrees.
- OK, that's 50 degrees.
- What is the measure of arc PQ?
- What is this arc right there?
- So we could use what we learned in the last problem.
- 50 degrees is an inscribed angle.
- So it's going to be half of the arc length measured in
- degrees, or arc angle, I guess you could say, of the arc that
- it intercepts.
- So this is 50, then this arc measure is
- going to be 100 degrees.
- So what we're trying to figure out, this arc measure right
- here, it's whatever you really have left over.
- Let's call this x.
- So we know that x plus 100 plus 140, well, that goes all
- the way around the circle, right?
- That's going to be equal to 360 degrees.
- So you have x plus 240 is equal to 360 degrees.
- Subtract 240 from both sides, and you get x is
- equal to 120 degrees.
- Choice D.
- Problem 78: The vertices of triangle ABC are A is 2 comma
- 1, B is 3 comma 4, and C is 1 comma 3.
- If triangle ABC is translated one unit down-- so one unit
- down means that we're subtracting 1 from the y's--
- and three units to the left-- so that means you're
- subtracting 3 from the x's-- to create triangle DEF, what
- are the coordinates of the vertices of DEF?
- So let's see, we have A is 2 comma 1.
- And that's going to be translated to point D.
- What do they say?
- One unit down.
- One unit down means the y decreases by 1.
- So the y-coordinate's going to be-- decrease that
- by 1, you get 0.
- And three units to the left, that means you decrease the
- x-coordinate by 3.
- So 2 minus 3 is negative one.
- Actually, just looking at the choices, we're already done.
- D is the only choice that had a negative 1 comma 0.
- But let's look at the other ones.
- Maybe somehow they changed the lettering or something.
- Let's see point B was 3 comma 4.
- So what do we do?
- With x, we said three units to the left, so that
- becomes point E.
- So you take this three units to left.
- You're subtracting 3 from there, that becomes 0.
- If you take the y-coordinate one unit down,
- that becomes a 3.
- So E is 0 comma 3.
- So that's still consistent with this.
- And let's just make sure that the F works.
- So C was 1 comma 3.
- If we take 3 away, we shift it to the left by 3, so that's
- taking 3 away from the x-coordinate, so
- that's minus 2.
- And if you're shifting the y-coordinate down one, that's
- taking 1 from it, so that's 3, so F is minus 2 comma 2.
- And so D was definitely the answer.
- Problem 79: If triangle ABC is rotated 180 degrees about the
- origin, what are the coordinates of A prime?
- OK, so we're going to rotate this thing 180 degrees.
- And essentially, we can worry about all the coordinates, but
- they just want to know where A prime is, so wherever A sits
- after we've rotated it.
- So let's think about it a little bit.
- When we've rotated the C, this is going to be the new C.
- It's going to be right there.
- B is going to be, when you rotate it-- I just want to
- make sure I'm doing this right.
- This is almost like a visualization problem.
- If you go four to the right and up three.
- So if you were to rotate it around.
- Let's say this little arrow I drew with the four, if I were
- to rotate that 180 degrees-- I just want to make sure I do it
- right-- then that's going to be four this way.
- So you're going to be at that point.
- And then instead of three up, you're going to go three down.
- So that's where A prime is going to be.
- Right at that point.
- And that's the point minus five comma minus four.
- And that's our answer.
- But let's figure out all of the other points.
- So B, you go up three and to the right two.
- The new B, you'd go three down to the left.
- OK, it would be there.
- So that's the new B.
- And so that looks right.
- That's what the triangle is going to look like.
- This is B prime, this is A prime, this is C prime.
- This is the new triangle when you rotate it 180 degrees.
- That was a challenging visualization problem, but the
- key thing is just to keep visualizing
- going around 180 degrees.
- So minus five comma minus four, that's
- choice A for A prime.
- Problem 80: Trapezoid ABCD below is to be translated to
- trapezoid A prime, B prime, C prime, D prime by the
- following motion rule.
- What will be the coordinates of vertex C prime?
- OK, so that's all we have to worry about.
- We're at C and we're going to translate it to some C prime.
- What of the coordinates of C?
- x is equal to five, and y is equal to 1.
- So what's C prime going to be equal to?
- C prime, we'll just use this mapping rule.
- You add 3 to the x, so it's going to be 8, and you
- subtract 4 from the y.
- 1 minus 4 is minus 3.
- 8 minus 3.
- That's choice D.
- And we're all done.
- We're all done with the entire California Standards geometry
- test. And frankly, I started off grumbling a little bit,
- because I thought they were getting a little bit too
- obsessed with the terminology.
- But overall, I think this was a pretty good test. And if you
- understood every problem on this exam, I think you know
- your geometry pretty well.
- Anyway, see you in the next video, although this is the
- last in this geometry series.
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