I saw this problem on the "Mind Your Decisions" YouTube channel and here are my two solutions, I haven't watched the video/
The Problem
In my sketch we have a rectangle of unknown dimensions, a quarter-circle inscribed in the rectangle, a semicircle positioned as shown with its centre on bottom of the rectangle, and a line of length 5 drawn from the corner of the rectangle and tangent to the semicircle. The challenge is to find the area of the rectangle.
At first sight this is impossible, we only have one distance so how can we find the area?
First Solution
When a problem involves a tangent to a circle or part of one, it almost always uses the fact the a tangent makes a 90 degree angle with the radius at the point of contact; and when a problem involves a right-angled triangle and we are interested in distances, that points to Pythagoras' Theorem.
So here's the diagram again: I have labelled some key points, called the radii of the quarter and semicircle r1 and r2 respectively, drawn in the radius to point of contact D, and marked the right angle.
Now we see we have a right-angles triangle with hypotenuse r1 + r2 and its other sides 5 and r2. We can apply Pythagoras and then use some algebra on the resulting equation as follows.
But r1 and r1 + r2 are precisely the height and length of the rectangle and their product is the area of the rectangle, which must therefore be 25. r1 and r2 can take different values as long as the satisfy the relationship we ended up with above and the area must always be 25.
Second solution
We are not told the values of r1 and r2, so it must not matter as such and we are free to choose them as we like as long as our choice is compatible with the given geometry.
Very well: let's set r2 = 0. Now the semicircle collapses to a point at B, the rectangle becomes a square, and the tangent degenerates into the line AB, r2 becomes 5 and the area is (r2)2 = 25.
This Problem Is Not Impossible
I saw this problem on the "Mind Your Decisions" YouTube channel and here are my two solutions, I haven't watched the video/
The Problem
In my sketch we have a rectangle of unknown dimensions, a quarter-circle inscribed in the rectangle, a semicircle positioned as shown with its centre on bottom of the rectangle, and a line of length 5 drawn from the corner of the rectangle and tangent to the semicircle. The challenge is to find the area of the rectangle.
At first sight this is impossible, we only have one distance so how can we find the area?
First Solution
When a problem involves a tangent to a circle or part of one, it almost always uses the fact the a tangent makes a 90 degree angle with the radius at the point of contact; and when a problem involves a right-angled triangle and we are interested in distances, that points to Pythagoras' Theorem.
So here's the diagram again: I have labelled some key points, called the radii of the quarter and semicircle r1 and r2 respectively, drawn in the radius to point of contact D, and marked the right angle.
Now we see we have a right-angles triangle with hypotenuse r1 + r2 and its other sides 5 and r2. We can apply Pythagoras and then use some algebra on the resulting equation as follows.
But r1 and r1 + r2 are precisely the height and length of the rectangle and their product is the area of the rectangle, which must therefore be 25. r1 and r2 can take different values as long as the satisfy the relationship we ended up with above and the area must always be 25.
Second solution
We are not told the values of r1 and r2, so it must not matter as such and we are free to choose them as we like as long as our choice is compatible with the given geometry.
Very well: let's set r2 = 0. Now the semicircle collapses to a point at B, the rectangle becomes a square, and the tangent degenerates into the line AB, r2 becomes 5 and the area is (r2)2 = 25.