GMAT Quantitative Section. Урок 59. Геометрические задачи

На тесте GMAT Вы можете встретиться с большим спектром всевозможных геометрических задач. Сегодня мы продолжим наш разговор и поговорим о задачах, в которых рассматривается окружность с вписанным или описанным квадратом. Такие задачи не очень сложны, но потребуют от Вас внимания.

There are lots of types of different geometry problems, which can appear on GMAT. Today we will talk about such problems again.

Circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center.

Area can be calculated using the formula S=АR^2, R is a radius of a circle.
Circumference can be calculated using the formula C=2АR, R is a radius of a circle.

Problems with circles often contain inscribed or circumscribed squares. Here you have to remember that  inscribed square has its side equal to 2R, R is a radius of a circle, and for the circumscribed square diameter of the circle is the diagonal of the square. Let’s look at some examples.


    Into a square with side K is inscribed a circle with radius r. If the ratio of area of square to the area of circle is P and the ratio of perimeter of the square to that of the circle is Q. Which of the following must be true?

(A) P/Q > 1
(B) P/Q = 1
(C) 1 > P/Q > 1/2
(D) P/Q = 1/2
(E) P/Q < 1/2

{B} Let’s take a Square with side K. A circle inscribed into such a square will have a radius of K/2, so,
r = K/2. Thus, area of the square is K^2 and area of the circle is А K^2/4. The ratio (P) is 4/ А.
The perimeter of a square is 4K and perimeter of the circle is АK. The ratio (Q) is 4/ А. Therefore, the ratio of P to Q is 1.
    A circle is inscribed in a square. If a diagonal running through the center of the circle is 4 cm long, what is the area of the square that is not occupied by the circle?

(A) 1.7
(B) 2.7
(C) 12
(D) 24
(E) 25

{A} If the diagonal of the square is 4 cm, then the sides are 4/2 = 2 Ч 2 based on the triangle rule.
So, Radius is 2 (as a half of a side). The area of the circle is 2pi and that of the square is 8.
8 – 2pi = 8 - 6.28  = 1.7 (approx)
    A square is inscribed in a circle of radius 4. What is the area of the square?

(A) 8
(B) 16
(C) 32
(D) 48
(E) 64

{C} The diameter of the circle is the diagonal of the square. So, diagonal of a square equals 8. The side of the square is 8/2 =  42, the area of the square is 32.

Resume:
This lesson shows some typical problems connected with the circles and inscribed or circumscribed square and the most effective ways to solve them in 2 minutes or even faster and add some points to your GMAT total score.

Material prepared by Ksenia Zueva, GMAT consultant at MBA Strategy