Problem number 20 states. The diagram shows three squares, PQRS, TRVU, and UWXY. They are placed together edge to edge. Points P, T, and X lie on the same straight line. The area of PQRS is 36, and the area of TRVU is 16. What is the area of triangle PXV? So this is the diagram that the problem gives us, and let's go over what we know. So we know the areas of the two squares on the left are 36 and 16, which means that they have side lengths of 6 and 4. At this point, it might be helpful to think of the points as coordinates. So we'll say that S is at 0, 0, so that all the coordinates lie in quadrant 1 and are positive. So R would be 6, 0, as it is 6 units to the right of R. And V would be 10, 0, as it is 4 units to the right of R. Doing this for every point, this is what we get. The problem tells us that P, T, and X lie on the same line, so let's solve for this line. So the equation for the line would follow y equals mx plus b. In this case, the y-intercept is 6 because the line crosses x equals 0 at y equals 6. Now let's calculate the slope. So we do this by doing change in y over change in x. So 4 minus 6 over 6 minus 0, so a slope of minus 1 third. So we get y equals minus 1 third x plus 6. But now we need to find the location of point x. So let's say that our square has width and height a. This means the coordinates of point x will be a more than 10 for the x-coordinate and a less than 4 for the y-coordinate. Since we know this point lies on the line, let's plug the x and y-coordinates into the line equation and solve for a. Let's distribute the minus 1 third and now let's add a to both sides and subtract 4. Now let's combine like terms and multiply both sides by 3, add 4 to both sides and divide both sides by 2 and we get that a equals 2. So the small square's side length is going to be equal to 2. So now we have all the information necessary to find the area of this triangle. You can use many techniques like Heron's formula or the shoelace method, but the way that I'm going to show you involves no knowledge of advanced formulas or theorems, just basic formulas for geometric shapes. So to find the area of the triangle, we will find the area of the yellow trapezoid and subtract the areas of the red triangles. So the area of the trapezoid will be the base times the average of its two heights and the area of the triangle is just one half base times height. It is worth noting that all the bases and heights will be different for each triangle and trapezoid. So plugging in the values, this is what we get. Simplifying it down, we get 12 times 4 minus 30 minus 2, which gives us an area of 16. So that's our final answer. So the question asked us, what is the area of triangle PXV? The answer is 16, letter C.

triangle area

squares

geometry

coordinates

trapezoid