Translate the triangle so that point 𝑂 moves to point 𝑃.
So in this question, we’re looking to translate. When you translate a shape, it moves its position. So it goes up, down, left, right. However, its orientation does not change. So therefore, it stays looking the same as it would now. So the translation we want to carry out needs to move point 𝑂 to point 𝑃. And therefore, all the other points or vertices on our triangle are going to have to follow the same translation. So to get from point 𝑂 to point 𝑃, first of all we could see that we go seven spaces right or seven squares right. And then, we go one square down. We could’ve gone the other way around. So we could’ve gone one down and seven right. And either of these would work the same way because it would change our position from 𝑂 to 𝑃.
The other way that we could’ve represented our translation was by using a vector. So we have something here called a column vector. So we have 𝑥 over 𝑦. It doesn’t have a line in between because it’s not a fraction. And well, the vector, the top number, tells us whether it’s left or right because it’s the 𝑥-direction. So negative is left. Positive is right. And the bottom number tells us if it’s up or down because this is in the 𝑦-direction, so positive up, negative down. So therefore, the vector for our translation would be seven, negative one. And that’s cause it goes seven to the right. And it goes one down.
Okay, so now we know what the translation is, we’d carry out the same translation on the other two vertices. So I’m gonna start with the bottom left vertex of the triangle. So to find out where the new vertex is going to be on our translated shape, what we’re gonna do is go one down or seven right or seven right and one down. And when we do that, we get to the point shown here in blue. So then, I’m gonna do the final vertex. And we’re gonna do the same. So this time, we’re gonna go seven right, one down. So when we do that, we get to the point shown here by the other blue cross. So we’ve now found the three vertices of our new triangle. So then, what I do is I draw in our new shape, so our new triangle. And we can see its orientation hasn’t changed. It hasn’t turned. Its size hasn’t changed. It has just changed position.
So we can double check as well by checking that the sides of the new shape are the same as the original. So we’ve both got the horizontal and vertical sides of four. So they’re four units long. And then we’ve also got the hypotenuse. Well, if they’re both four units long, then the hypotenuse can be the same on each of our triangles. So there we go. We’ve translated the triangle. So that point 𝑂 moves to point 𝑃. And that’s part a) complete. We now have a part b) that we’re going to move on to.
So in part b), what we’re looking to do is reflect the triangle so that the point 𝑄 moves to point 𝑅.
So if we’re looking to reflect our triangle, what we’re gonna need to find is a mirror line because for a reflection to take place, there has to be a line of reflection or mirror line. Now for 𝑄 to move to point 𝑅 with our reflection, then what we’re gonna have to do is look at the line 𝑄𝑅. And if we’ve got the line 𝑄𝑅, then our mirror line is going to bisect this. So it’s going to be at the midpoint.
Well, if we count the spaces, we can see that the line 𝑄𝑅 is six diagonals long. So it’s six diagonals of our square’s long. So therefore, if we want the bisector of this or the halfway point, then it’s going to be three diagonals from either end. So I’ve marked this with a pink cross. And this is in fact going to be the midpoint or bisector of the line. So therefore, we need to draw that bisector because that bisector is going to be our mirror line. So now, what I’ve done is drawn on our mirror line or line of reflection. And as I said, this is the bisector of the segment 𝑄𝑅.
So now what we need to do is find out where the other vertices of our new triangle are going to be. Well, first of all, we can see that the line 𝑄𝑅 is perpendicular to our reflection line. And the construction lines for our other two vertices have to also be perpendicular to our mirror line. So first of all, I’m gonna start with this vertex here, the middle vertex. And I can see that this is one diagonal square away from our mirror line if we’re looking at a perpendicular construction line. So therefore, the vertex on our reflected shape will have to mirror this. So it’s gonna be one diagonal away from our mirror line. And again, it’s gonna be perpendicular. So I’ve marked where the vertex will be with a cross.
So now if I look at the final vertex, we can see again this one is three diagonals away from our mirror line. So therefore, the vertex of our reflected shape is going to have to also be the same distance away from the mirror line. So I’ve shown that here because you can see that it’s three squares diagonally away from the mirror line. And once again, the line needs to be perpendicular as the others all are. So now that we’ve added in all the sides to our triangle, we can say that the reflected shape here has the same size sides as the original. Which is what we’re looking for. And we can see that it is a direct reflection across the mirror line that we’ve shown. So therefore, we can say that we have reflected the triangle so that point 𝑄 moves to point 𝑅.