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Pi Han Goh, Mahindra Jain, Gautam Jha, and 3 others

• A Former Brilliant Member
• Calvin Lin
• Jimin Khim

contributed

You should be familiar with Congruent and Similar Triangles.

In mathematics, we say that two objects are similar if they have the same shape, but not necessarily the same size. This means that we can obtain one figure from the other through a process of expansion or contraction, possibly followed by translation, rotation or reflection. If the objects also have the same size, then they are congruent.

Contents

• Identify Similar Polygons
• Similar Polygons - Area and Perimeter Relations
• Similar Polygons Problem Solving - Basic

For a general shape, it can be tricky to show that two items are similar. We have to check all corresponding angles and ratios of side lengths before we can reach a conclusion. For example, a \( 1 \times 2 \) rectangle is not similar to a \( 2 \times 3 \) rectangle, even though they both have 4 right angles, since their side lengths have different ratios. Likewise, a square of side length 1 is not similar with a rhombus of side length 1 and vertex angle \( 60 ^ \circ \), even through they all have side lengths of 1.

For triangles, due to the cosine rule and sine rule, we have much more options of determining if they are congruent or similar triangles.

Show that all circles are similar to each other.

If we lay the circles over each other such that their centers coincide, then the expansion by factor \( \frac{R_2} {R_1} \) \((\)where \( R_1\) and \(R_2\) are the radii of the two circles, respectively\()\) maps the first circle to the second. \(_\square\)

Note: In a similar fashion, we can show that all regular polygons with the same number of edges are similar to each other. Why doesn't this work for rectangles?

The following result holds for similar figures \(F_1\) and \(F_2\) with a corresponding side length ratio of \(R:\)

• The ratio of their perimeters and diagonals will be in the same ratio \(R\).

• The ratio of their areas will be \( R^2 \).

If the sides of two rectangles are in the ratio \( 3 : 5 \), what is the ratio of their areas?

The ratio of their areas would be \( 3^2 : 5^2 = 9 : 25 \). \(_\square\)

1% 1.01% 2.01% 3.01% Reveal the answer

If the sides of a square are increased by 1%, then by what percent will the area of this square be increased?

The correct answer is: 2.01%

Its surface area increased 18 times Its surface area increased 27 times Its surface area remained the same Its surface area increased 9 times Reveal the answer

A spherical balloon is inflated until its volume becomes 27 times its original volume. Which of the following is true?

The correct answer is: Its surface area increased 9 times

Both occupy the same percentage of area No Depends on the size of pizzas and boxes Yes Reveal the answer

Usually, pizzas are circular and they come in boxes with a square base. The circular pizza covers some percentage of area of the square base of the box. Let's say someone makes a square pizza and puts it in a box with a circular base. Will the square pizza cover more percentage of area in the box with circular base than the circular pizza in the box with square base?

Assume that both the pizzas fit perfectly inside their respective boxes.

The correct answer is: No

If the area of a square is increased 21% percent, what is the increase in the perimeter of the same square (in percentage)?

Let \(L\) denote the original length of the square and let \(L'\) denote the new length of the square after the area has been increased. Then, \( \frac{ L'^2}{L^2} = 1.21 = 1.10^2 \). So the length \(L\) was increased by 10%. Since the perimeter of the square is proportional to its side length, the the perimeter of the square also increased by 10%. \(_\square\)

Reveal the answer

If all of the rectangles in the figure are squares and the area of the shaded square is 25, find the area of square ABCD.

The correct answer is: 100

Reveal the answer

\(ABCD\) is a square with area \( 720 \). Points \(E, F, G, H \) trisect the sides of the square, such that \( \frac{ AE}{EB} = \frac{ BF}{FC} = \frac{ CG}{GD} = \frac{DH}{HA} = 2 \). What is the area of square \( EFGH \)?

The correct answer is: 400

Let the side length of square \(ABCD\) be \(3s\). Then, the side length of square \( EFGH\) is \( \sqrt{ s^2 + (2s)^2 } = \sqrt{5} s \). Hence, the ratio of their area is \( \left( \frac{ \sqrt{5}s } {3s} \right) ^2 = \frac{5}{9} \). Thus, the area of \(EFGH\) is \( \frac{5}{9} \times 720 = 400 \).

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