Grades 11-12 Video Solutions 2025-2025_11-12

Video Transcription

Hello, this is problem 14 on the grades 11 and 12, 2025 Math Kangaroo Contest.  So, the diagram shows a quarter circle, OPQ, and a triangle, OPR, where the length of OP is r, i.e. the radius of the circle is r.  The two shaded regions have the same area. What is the length of OR? Our answer choices are a, pi r over 2, b, 3r over 2, c, pi r, d, 2 over pi, and e, pi over 2r.  So that's kind of a lot.  And initially, this problem might not be super clear, because these two shapes are really, really weird shapes, and we don't really have any direct formulae to compute them.  So, you know, that's kind of unfortunate.  But what if, instead, we didn't pay attention to just the gray areas, we paid attention to this, like, gray area, plus this, like, unshaded area that I'm now shading in in green.  What we do know is that, like, this area that is now covered in, you know, both blue and green, that's sort of in common between both of the shapes.  So we have that, and let's call this point maybe M, because it's in sort of the middle of everything.  And we know that the, like, P, M, Q, O is shared between, like, triangle P, O, R, and quarter circle O, P, Q.  And if we subtract this area off, then we're left with the equality between R, Q, M equals, you know, the arc P, M.  So we're just, like, adding this area back in, and we still, like, preserve our equality, but we just get shapes that are easier to work with, because we know how to find the area of a quarter circle, and we also know how to find the area of a, you know, triangle.  So then, let's just, like, actually compare those areas. We get that, let's do the triangle first. R times the height, which, you know, is the value we're looking for.  So R times height over 2 is equal to R squared times pi all over 4, because we're, you know, working with a quarter circle.  So we find the area of the entire circle, and, you know, divide it by 4. And this just becomes kind of an algebra problem.  So I'm multiplying this side by 2 over R, and this side by 2 over R to cancel the 2, cancel the 2, cancel the R, cancel the R.  We're left with H is equal to R squared times pi times 2 over 4 times R. Cancelling, we obtain R times pi, and then we get a 2 on the denominator.  So we get that H is equal to R times pi all over 2, which is the same as answer choice A. So our answer is A.

Video Summary

In this Math Kangaroo problem for grades 11-12, the challenge is to find the length of OR given a quarter circle OPQ and a triangle OPR with equal shaded areas. By focusing on combining unshaded and shaded areas, the problem simplifies to comparing areas of known shapes: a quarter circle and a triangle. The calculations involve equating the triangle's area \( R \times \text{height} / 2 \) to that of the quarter circle \( R^2 \pi / 4 \). Solving this results in the formula \( H = R \pi / 2 \), corresponding to answer choice A: \( \frac{\pi r}{2} \).

Keywords

Math Kangaroo

quarter circle

triangle area

equating areas

formula derivation