GRE geometry strategies

Draw the figure. If the problem gives a diagram, redraw it on your scratch paper with every label. If it does not give a diagram, draw one from the description. Label every known value, every variable, and every relationship the problem states. A clean, labeled diagram turns a word problem into a visual one, and the answer often becomes obvious once you can see what the problem is asking.

Triangle area: $\frac{1}{2} \times \text{base} \times \text{height}$. Circle area: $\pi r^2$. Circle circumference: $2\pi r$. Rectangle area: $\text{length} \times \text{width}$. Pythagorean theorem for right triangles: $a^2 + b^2 = c^2$. Pythagorean triples to recognize instantly: 3-4-5, 5-12-13, and their multiples. Volume of a rectangular solid: $\text{length} \times \text{width} \times \text{height}$.

A right triangle has legs of length 3 and 4. What is the area of the circle that passes through all three vertices (the circumscribed circle)? Step 1: Draw the figure. Label the legs 3 and 4, and mark the right angle. Step 2: Find the hypotenuse. By the Pythagorean theorem: $3^2 + 4^2 = 9 + 16 = 25$, so the hypotenuse is 5. Step 3: Use the property. For a right triangle, the hypotenuse is the diameter of the circumscribed circle. So the diameter is 5, and the radius is $\frac{5}{2} = 2.5$. Step 4: Compute the area. $\pi r^2 = \pi (2.5)^2 = \pi \times 6.25 = 6.25\pi$. The answer is $6.25\pi$.