Video: SAT Practice Test 1 • Section 3 • Question 13

In the 𝑥𝑦-plane, an isosceles triangle has two vertices on the 𝑥-axis and one vertex on the 𝑦-axis at (0, 8). The perimeter of the triangle is 32 units. Which of the following could be another vertex of the triangle? [A] (0, 6) [B] (8, 0) [C] (0, −8) [D] (−6, 0)


Video Transcript

In the 𝑥𝑦-plane, an isosceles triangle has two vertices on the 𝑥-axis and one vertex on the 𝑦-axis at zero, eight. The perimeter of the triangle is 32 units. Which of the following could be another vertex of the triangle? A) Zero, six, B) eight, zero, C) zero, negative eight, or D) negative six, zero.

So in this question, we’re told that we got an isosceles triangle. And one of the vertices is on the 𝑦-axis at zero, eight. And I’ve shown that here on my sketch. We’re told that the other two vertices are on the 𝑦-axis. So what does that tell us? Does that help us rule out any of the possible answers? Well, it does. Because if we know that the other two vertices are on the 𝑥-axis, then we know the coordinates because the coordinates are going to be something because the 𝑥-value will change. But the second coordinate or the 𝑦-coordinate will be zero. And that’s because at the 𝑥-axis, the 𝑦-coordinates are all zero. And that’s because the 𝑥-axis could also be called a line, whose equation is 𝑦 equals zero.

Well, because of this, we can rule out a couple of answers. So first of all, we can rule out A. And that’s because A has zero as its 𝑥-coordinate. However, the 𝑦-coordinate is six. And the next answer we can rule out is C. And that’s because that’s zero, negative eight. So the 𝑦-value is negative eight. So great, we’ve ruled two out. We just need to now decide whether it’s answer B which is eight, zero or answer D which is negative six, zero.

Well, we’re gonna take a look at B first. And I’ve marked on eight, zero on our 𝑥-axis. So we now have the point zero, eight and the point eight, zero. Well, if I join them together with a line, then this line is going to be part of the perimeter of our triangle. And we know that the perimeter of the triangle is 32 units. So let’s calculate the length of this line to see if this can actually help us work out whether B is correct or incorrect.

Well, to work out the length of the line, we have a right-angled triangle, whose height is eight units and whose width is eight units. So therefore, what we can use is the Pythagorean theorem. And what the Pythagorean theorem tells us is that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. And this is where 𝑐 squared is the longest side or the hypotenuse. And this is found opposite the right angle.

So if we look at this in our diagram, we’re gonna have 𝑥 squared. And that’s because 𝑥 is our 𝑐; it’s our hypotenuse. It’s equal to eight squared plus eight squared cause these are the two sides of our right-angled triangle. So therefore, 𝑥 squared is gonna be equal 128. And that’s because eight squared is 64. 64 add 64 is 128. Well, now to find 𝑥, what we can do is take the square root of each side of the equation. And when do that, we get 𝑥 is equal to 11.304 et cetera. So that gives us a decimal answer.

Well, as we can see that the perimeter is exactly 32 units, then if this is one of our sides, this is not gonna add up to give us exactly 32 units for our perimeter. For example, if we put the other vertex at negative eight, zero so that we had our two equal sides because it was an isosceles triangle as the two sides that went from negative eight, zero to zero, eight and eight, zero to zero, eight, then that means we’d have two sides, each of the length of 11.30 et cetera and one side of 16. Well, if you add these together, we don’t get 32 cause 16 add 11 would give us 27 add another 11 would give us 38. And then, you also got the .307 et cetera and two of those to add together. So it’s not gonna give us the 32 units you want.

There are other ways. We could use lines to make an isosceles, given the 𝑥-line that we already found. But again, none of these will add up to give an answer that is 32 units exactly. So we can rule out B. But that means D is gonna be the correct answer. But I don’t want to just accept that. What I want to do is show it, show that it is the correct answer.

So now if we look at D, we’ve got negative six, zero as one of our vertices. So I’ve marked this on and I’ve drawn the line. So what we want to do is the same process, this time find the length of that line. Again, we can use the Pythagorean theorem because we got a right-angled triangle. And again, I’ve called the hypotenuse 𝑥.

So we get 𝑥 squared is equal to eight squared plus six squared. Well, 𝑥 squared is gonna be equal to 100. And that’s because eight times eight is 64 and 64 add 36 because six times six is 36 is 100. And then if we square root each side, we’re gonna get 𝑥 is equal to 10. And therefore, our hypotenuse is equal to 10. So the line between negative six, zero and zero, eight is 10.

So if we did the same to the other side and we made a point at six, zero, we’d now have our isosceles triangle. We’d have two sides that will length 10. And the final side which was length 12 because from negative six to zero is six and from zero to six is six. Add them together, you get 12. So therefore, if we calculated the perimeter, it’ll be 10 add 10 add 12 which will be equal to 32. So it will be 32 units, which is what we’re looking for.

So therefore, the correct answer is D, negative six, zero.

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