## DS Geometry Questions

### Transcript

The Geometry Questions in Data Sufficiency. In this lesson I will assume that you are already seen the geometry lesson. So in other words you are familiar with the basic facts of geometry. So we're not gonna spend any time reviewing those. We'll just assume that those are familiar and we'll talk about how these apply in Data Sufficiency.

You see Geometry in Data Sufficiency is an interesting case for a few reasons. In particular, the DS Geometry Questions are a rigorous test of your visual skills in your capacity for you, for imagination. That is your ability to imagine the different kinds of shapes that would be consistent with a particular given set of constraints. That's ultimately what these questions are testing.

So the first really big idea is, don't trust diagrams. Don't naively believe the diagrams on the data sufficiency. If there is geometric information given in the prompt, then that, then the diagram will reflect the prompt information. But the diagram will not necessarily reflect the geometric information given in the statements.

It may will be that information given in the statement changes the diagram in some significant way. That's exactly what you have to decide, in fact. More importantly, diagrams will often suggest a special relationship that is not guaranteed by any of the information given. This is really tricky because even more so than in problem solving, the data sufficiency diagrams are designed in part to deceive you.

That is to say to illicit assumptions that are unjustified. So for example, if you were to see this, you would certainly be tempted to make an assumption. That C is a right angle, and if C were a right angle, then you could use the Pythagorean Theorem, AC would be three. You'd be very tempted to draw those conclusions.

But we don't know that. Nothing is actually guaranteeing that. the angle is drawn to look like a right angle, so even on the problem solving question, where things are drawn to scale, we'd be guaranteed that the angle. Is close to 90, we wouldn't be guaranteed that it was exactly a 90 degree angle, even on, on problem solving.

But on data sufficiency we're not even guaranteed that it's drawn to scale. All bets are off in Data Sufficiency. That drawing could suggest one of these shapes, so here C is an obtuse angle here C. Is a wildly acute angle. There could be a right angle at C.

There could be a right angle at B. There could be a very different triangle all together. We don't know. We're not guaranteed anything and we can't make assumptions. So for example if you were to see that particular shape you might be very tempted to assume that the shape is a square.

Now remember back in Geometry we talked about how the square is a highly elite shape. If you know something is square, you know a ton. If you don't know that something is a square, it's very hard to prove that it's a square. If the prompt or statement tells us that it's a square, well then that the jackpot.

Then we have a ton of information. That's great. But in the absence of such guarantees it's very hard to prove that the shape has the unique combination, the special geometric qualities that constitutes a square. Very hard to prove that something is a square.

So here's a practice problem. Pause the video and then we'll talk about this. So right away we should be suspicious of this because we're not told that the shape is a square, we're asked is it a square. Essentially we're being asked do we have enough information to prove that something is a square.

And of course we should keep in mind that it's very hard to prove that something is a square. So you might have been tempted to assume that one or both of the statements were enough to guarantee a yes answer. This is a really tricky thing. If we knew that the shape were square, then both statement (1) and statement (2) would be true.

But that' s not what data sufficiency is asking, it's not asking if we know the answer to the prompt then can we decide whether the statements would be true. No it works the other way around. Given that we know the statements can we answer the prompt question. That's what we're trying to do and unfortunately we don't know that that's a square.

We don't have enough information. So, in fact, even if we had the combined information, the shape might look something like this thing at the bottom. So notice, we have two right angles, one at E, one at G. We have length EF equals length EH. So those two lengths are congruent.

But that irregular quadrilateral is not even pretending to be a square. So this shape could be a square or it could be something very different, something that is not obeying any special properties at all. So, combine the statements are insufficient, that means that the answer is (E).

Okay, the second big idea, is that we need to be given a length to find a length. What would I mean by this? In a DS Geometry value question, in which we were asked to find a certain length. We need to be given the measurement of another particular length. In other words, if we're not given a length, there's no way to find a length. That's a very important idea.

If all we're given is angle measurements or a ratio information, the statement that two lengths are equal or one is twice the length of the other, something along those lines-. Ratio information and angle information is enough to guarantee that the shape could be scaled up or down to any similar figure, but no particular length in the figure would be locked in place.

For example, we're told that angle K is 140 degree. And that KL = 2*(JK)... So we're given angle information and ratio information. That alone is not enough to fix any length because the figure could be scaled up or scaled down to any similar shape. So yes it would be the same basic shape but we could make a tiny copy, we could make a huge copy and that means that.

No length specified. Without one length measurement specified to lock the figure in shape, lock it in place, all the lengths could be scaled up or down arbitrarily, and there'd be no way to find any particular length in the figure. The third big idea, this is a hard one. User your visual imagination to see variants.

This takes some practice. It may help you to play with shapes to see possible variants. So, first of all, just on a very practical level, I'll say notice that two pieces of information, say (angle + length, or two lengths) is never enough to determine a unique triangle. But, three pieces of information, (2 angles + a length, two lengths + an angle, three lengths), combinations like this are almost always enough to, to determine fully a unique triangle.

And, again, it's worthwhile to play around with this a little bit. For example, take straws and cut them into different lengths. You have different length segments. You could take anything long and thin, pipe cleaners, anything like that. And build triangles, and actually play around. What happens if you have two fixed lengths?

What are possible lengths for the third length? If you have a. One fixed length and one fixed angle, how does that work if you have two fixed angles, how do these things lock things in place, what can change, what can't change. These are the things to explore, and the more you can play around physically, with actually looking at shapes, the more you will develop your, your sense.

Of visual imagination and, and the visual possibilities of the different shapes. So here's a practice question. Pause the video and then we'll talk about this. So in the diagram, is triangle ABC an equilateral triangle? So notice that's something quite elite.

It's like proving something is a square to prove that is an equilateral triangle. It's hard to do. So far, we're guaranteed that we have a right angle at D. So, BD is an altitude. It makes a 90 degree angle with the base. Statement (1) gives us one angle.

It gives the angle at C. So from that we know that the right half triangle, triangle BCD, that has to be a 30-60-90 triangle. But we know nothing about the triangle on the left, triangle ABD. That point A, we could slide that in or out. It, it could be another 60 degree angle so we could have an equilateral triangle, but it could be slid out, in or out to be almost any other acute angle.

So we actually know nothing about the left hand, left side triangle, so we have no way to determine whether or not the triangle is equilateral. So this first statement by itself is insufficient. Now we have to forget about that entirely. We don't know anything about angle measurements. Just look at statement (2).

For statement (2), we know BD would be an altitude that intersects the midpoint of the base. So we know that ABC must be isosceles, with leg AB equal to leg BC. So we know that that's true. And so again, it could be equilateral or it could be just another isosceles triangle.

So the angle A and the angle C would be equal to each other, but there's no guarantee that they have to be 60 degrees they could be anything. So this by itself is also insufficient. Well, both statements individually were insufficient, and now we have to combine the information. Well, we know from statement (2) that we have an isosceles triangle.

And then from statement (1), if we're given a 60 degree angle, well, any isosceles triangle with one 60 degree angle. Must be equilateral. So with both pieces of information together we can determine yes, it is an isosceles triangle. We get a definitive answer.

Because we have a definitive answer, that means we have sufficient information. So together there not, sorry separately they were not sufficient but together they are sufficient. So that's an answer of (C). And again, to build you're visual imagination skills you have to practice.

For any set of geometric constraints, practice drawing as many different shapes as possible consistent with that combination of constraints. When you are working on a Data Sufficiency Geometry Question, draw whatever shapes are possible with each statement individually, then, it maybe, what shapes are possible with the statement together? In summary.

Remember, do not trust the DS geometric diagrams. Don't be misled to assume things that are not guaranteed, that's very important. Remember the need to get a length in order to find a length. And finally, practice your visual imagination skills. Try to visualize the different shapes possible within any set of constraints.

Read full transcriptYou see Geometry in Data Sufficiency is an interesting case for a few reasons. In particular, the DS Geometry Questions are a rigorous test of your visual skills in your capacity for you, for imagination. That is your ability to imagine the different kinds of shapes that would be consistent with a particular given set of constraints. That's ultimately what these questions are testing.

So the first really big idea is, don't trust diagrams. Don't naively believe the diagrams on the data sufficiency. If there is geometric information given in the prompt, then that, then the diagram will reflect the prompt information. But the diagram will not necessarily reflect the geometric information given in the statements.

It may will be that information given in the statement changes the diagram in some significant way. That's exactly what you have to decide, in fact. More importantly, diagrams will often suggest a special relationship that is not guaranteed by any of the information given. This is really tricky because even more so than in problem solving, the data sufficiency diagrams are designed in part to deceive you.

That is to say to illicit assumptions that are unjustified. So for example, if you were to see this, you would certainly be tempted to make an assumption. That C is a right angle, and if C were a right angle, then you could use the Pythagorean Theorem, AC would be three. You'd be very tempted to draw those conclusions.

But we don't know that. Nothing is actually guaranteeing that. the angle is drawn to look like a right angle, so even on the problem solving question, where things are drawn to scale, we'd be guaranteed that the angle. Is close to 90, we wouldn't be guaranteed that it was exactly a 90 degree angle, even on, on problem solving.

But on data sufficiency we're not even guaranteed that it's drawn to scale. All bets are off in Data Sufficiency. That drawing could suggest one of these shapes, so here C is an obtuse angle here C. Is a wildly acute angle. There could be a right angle at C.

There could be a right angle at B. There could be a very different triangle all together. We don't know. We're not guaranteed anything and we can't make assumptions. So for example if you were to see that particular shape you might be very tempted to assume that the shape is a square.

Now remember back in Geometry we talked about how the square is a highly elite shape. If you know something is square, you know a ton. If you don't know that something is a square, it's very hard to prove that it's a square. If the prompt or statement tells us that it's a square, well then that the jackpot.

Then we have a ton of information. That's great. But in the absence of such guarantees it's very hard to prove that the shape has the unique combination, the special geometric qualities that constitutes a square. Very hard to prove that something is a square.

So here's a practice problem. Pause the video and then we'll talk about this. So right away we should be suspicious of this because we're not told that the shape is a square, we're asked is it a square. Essentially we're being asked do we have enough information to prove that something is a square.

And of course we should keep in mind that it's very hard to prove that something is a square. So you might have been tempted to assume that one or both of the statements were enough to guarantee a yes answer. This is a really tricky thing. If we knew that the shape were square, then both statement (1) and statement (2) would be true.

But that' s not what data sufficiency is asking, it's not asking if we know the answer to the prompt then can we decide whether the statements would be true. No it works the other way around. Given that we know the statements can we answer the prompt question. That's what we're trying to do and unfortunately we don't know that that's a square.

We don't have enough information. So, in fact, even if we had the combined information, the shape might look something like this thing at the bottom. So notice, we have two right angles, one at E, one at G. We have length EF equals length EH. So those two lengths are congruent.

But that irregular quadrilateral is not even pretending to be a square. So this shape could be a square or it could be something very different, something that is not obeying any special properties at all. So, combine the statements are insufficient, that means that the answer is (E).

Okay, the second big idea, is that we need to be given a length to find a length. What would I mean by this? In a DS Geometry value question, in which we were asked to find a certain length. We need to be given the measurement of another particular length. In other words, if we're not given a length, there's no way to find a length. That's a very important idea.

If all we're given is angle measurements or a ratio information, the statement that two lengths are equal or one is twice the length of the other, something along those lines-. Ratio information and angle information is enough to guarantee that the shape could be scaled up or down to any similar figure, but no particular length in the figure would be locked in place.

For example, we're told that angle K is 140 degree. And that KL = 2*(JK)... So we're given angle information and ratio information. That alone is not enough to fix any length because the figure could be scaled up or scaled down to any similar shape. So yes it would be the same basic shape but we could make a tiny copy, we could make a huge copy and that means that.

No length specified. Without one length measurement specified to lock the figure in shape, lock it in place, all the lengths could be scaled up or down arbitrarily, and there'd be no way to find any particular length in the figure. The third big idea, this is a hard one. User your visual imagination to see variants.

This takes some practice. It may help you to play with shapes to see possible variants. So, first of all, just on a very practical level, I'll say notice that two pieces of information, say (angle + length, or two lengths) is never enough to determine a unique triangle. But, three pieces of information, (2 angles + a length, two lengths + an angle, three lengths), combinations like this are almost always enough to, to determine fully a unique triangle.

And, again, it's worthwhile to play around with this a little bit. For example, take straws and cut them into different lengths. You have different length segments. You could take anything long and thin, pipe cleaners, anything like that. And build triangles, and actually play around. What happens if you have two fixed lengths?

What are possible lengths for the third length? If you have a. One fixed length and one fixed angle, how does that work if you have two fixed angles, how do these things lock things in place, what can change, what can't change. These are the things to explore, and the more you can play around physically, with actually looking at shapes, the more you will develop your, your sense.

Of visual imagination and, and the visual possibilities of the different shapes. So here's a practice question. Pause the video and then we'll talk about this. So in the diagram, is triangle ABC an equilateral triangle? So notice that's something quite elite.

It's like proving something is a square to prove that is an equilateral triangle. It's hard to do. So far, we're guaranteed that we have a right angle at D. So, BD is an altitude. It makes a 90 degree angle with the base. Statement (1) gives us one angle.

It gives the angle at C. So from that we know that the right half triangle, triangle BCD, that has to be a 30-60-90 triangle. But we know nothing about the triangle on the left, triangle ABD. That point A, we could slide that in or out. It, it could be another 60 degree angle so we could have an equilateral triangle, but it could be slid out, in or out to be almost any other acute angle.

So we actually know nothing about the left hand, left side triangle, so we have no way to determine whether or not the triangle is equilateral. So this first statement by itself is insufficient. Now we have to forget about that entirely. We don't know anything about angle measurements. Just look at statement (2).

For statement (2), we know BD would be an altitude that intersects the midpoint of the base. So we know that ABC must be isosceles, with leg AB equal to leg BC. So we know that that's true. And so again, it could be equilateral or it could be just another isosceles triangle.

So the angle A and the angle C would be equal to each other, but there's no guarantee that they have to be 60 degrees they could be anything. So this by itself is also insufficient. Well, both statements individually were insufficient, and now we have to combine the information. Well, we know from statement (2) that we have an isosceles triangle.

And then from statement (1), if we're given a 60 degree angle, well, any isosceles triangle with one 60 degree angle. Must be equilateral. So with both pieces of information together we can determine yes, it is an isosceles triangle. We get a definitive answer.

Because we have a definitive answer, that means we have sufficient information. So together there not, sorry separately they were not sufficient but together they are sufficient. So that's an answer of (C). And again, to build you're visual imagination skills you have to practice.

For any set of geometric constraints, practice drawing as many different shapes as possible consistent with that combination of constraints. When you are working on a Data Sufficiency Geometry Question, draw whatever shapes are possible with each statement individually, then, it maybe, what shapes are possible with the statement together? In summary.

Remember, do not trust the DS geometric diagrams. Don't be misled to assume things that are not guaranteed, that's very important. Remember the need to get a length in order to find a length. And finally, practice your visual imagination skills. Try to visualize the different shapes possible within any set of constraints.