In the figure above, the radius of the circle with center
O is 1 and
BC = 1. What is the area of triangular region
ABC?
What’s your first step? Let’s use this problem as an opportunity to practice the Quant Process.
- Look at the problem and jot down the given information.
- Go through the information and organize all the known theorems in your mind that can be used.
- Work on the problem by using the theorems and formulas in the required order.
At a glance, you can see that the problem provides a diagram.
Draw! Make it big enough that you can add labels as you calculate new pieces of information and jot down any information given in the problem.
Before you dive in and try to find this height, ask yourself whether there are other possible ways to move forward. Sometimes, the “obvious” way turns out not to be the easiest way to proceed.
In particular, this is a “multi-shape” problem: you were given both a triangle and a circle.
Why did they include the circle? Pay particular attention to where the two shapes overlap.
The hypotenuse of the triangle is also a diameter of the circle. How can you use that to solve?
It turns out that when a triangle is inscribed in a circle (the 3 vertices of the triangle all sit on the circle), and the hypotenuse of that triangle is also a diameter of the circle, then the triangle in question is a right triangle.
In this case, the right angle is labeled
B. Is that information useful ? Well, if you’re trying to find the area of a right triangle, then you just need to know the lengths of the two legs:
AB and
BC. The problem says that
BC = 1, so the only unknown is
AB.
Now you have a choice: do you think it’ll be easier to find the length of
AB or to find the length of the vertical line that you drew in below point
B?
Because
ABC is a right triangle, it’s easier to find
AB. The short leg is 1 and the hypotenuse is 2.
Can you identify any theorem parameters in the given measurements? These match the 30-60-90 triangle parameters.
The length of
AB is
. Plug this into the area formula:
The correct answer is (B).
Key Takeaways for Multi-Shape Geometry.
(1) Examine the “overlap” between the shapes. Most likely, some rule about that connection exists and this rule will help make the problem easier to solve.
(2) Draw! This is key for any geometry problem, but especially so for multi-shape problems. There are too many moving parts; you need to keep track of everything in a clear way.
(3) Remember Quant Process: Reflect before you Work! In this case, the first, more obvious path would have been a lot more difficult to execute. Reflecting for a moment allowed you to notice the connection between the circle and the triangle. The subsequent solution path turned out to be much more straightforward.
