What is the perimeter of the triangle OPQ? (1) The area of triangle O

Breaking Down the Info:

Note point P is "adjustable"; there are many triangles we can draw that have two right angles and a height of 5.

Statement 1 Alone:

This tells us the base OQ is 20. Recall we need angle P to be a right angle. Intuitively, if you put the height (5) right in the middle of OQ, we will have an obtuse angle for P instead of a right angle, so you might see by moving the height around, there are only two symmetric cases where we have a right angle for P. Hence there is only one dimension where we have everything set up correctly. Sufficient.

To go more in-depth, set PH = 5 as the height. and set OH = x. We may write \(OP = \sqrt{x^2+25}\) from Pythagorean theorem on triangle OHP. Then from similar triangles, we have \(\frac{OH}{OP} = \frac{OP}{OQ}\), which is \(20x = x^2 + 25\).

The -20 on the b constant tells us solving this gives two solutions that add up to 20, hence the solutions are x and 20-x, so we'll get two solutions that give effectively the same triangle.

Statement 2 Alone:

This is the same as above. Sufficient.

Answer: D