Problem 18 - Complex numbers from A to...Z

Question

Let \(A B C D\) be a square with center \(O\) and let \(M, N\) be the midpoints of segments \(B O, C D\) respectively. Prove that triangle AMN is isosceles and right-angled.

Step-by-Step Solution

Answer

Based on the given information and the step-by-step solution, answer the following question: Question: Prove that triangle AMN is an isosceles and right-angled triangle, given a square ABCD with center O, and points M and N are the midpoints of segments BO and CD, respectively. Answer: To prove that triangle AMN is an isosceles and right-angled triangle, we first demonstrate that AM = AN by showing both lengths are the sum of a side length of the square and half the diagonal. These equal side lengths indicate that the triangle is isosceles. Secondly, we determine that angle MON is a right angle (90 degrees) through angle calculations in square ABCD. This proves that triangle AMN is a right-angled triangle. Thus, triangle AMN is both isosceles and right-angled.

Step 1: Draw the diagram

First, draw a diagram with square ABCD, center O, and midpoint M and N of segments BO and CD, respectively. This will make it easier to visualize the relationships between the points, angles, and side lengths.

Step 2: Determine midpoint properties

Since M and N are midpoints of segments BO and CD, respectively, recall that a midpoint divides a line segment into two equal parts. Thus, BM = MO and CN = ND.

Step 3: Take notice of right angles in square ABCD

Observe that square ABCD has four right angles. Specifically, angles A, B, C, and D are all right angles since they are interior angles of a square.

Step 4: Calculate angle MON

Notice that angle MON lies within the square, connect points M, O, and N together. To find the measure of angle MON, note that since angle BOC is a right angle (due to it being an angle in square ABCD), and angle MOB and angle NOC are both half of angle BOC due to the definition of midpoints (see Step 2): \( \angle MON = 180^\circ - (\angle MOB + \angle NOC) = 180^\circ - (45^\circ+45^\circ ) = 90^\circ \) Thus, angle MON is a right angle.

Step 5: Apply the Pythagorean theorem to triangle AMN

Consider the triangle AMN. To find the side lengths, notice that AM = AB + BM and AN = AD + DN. Using the Pythagorean theorem, we can compute the length of MN: \(MN^2 = AM^2 - AN^2\) Since AM and AN have the same length (both being side lengths of a square plus half the diagonal of the square), we can conclude that MN is equal to half the square's diagonal. Thus, AM = AN.

Step 6: Conclude that triangle AMN is isosceles and right-angled

We can now conclude that triangle AMN is an isosceles triangle, as AM = AN (proven in Step 5). Furthermore, with angle MON being a right angle (proven in Step 4), triangle AMN is also a right-angled triangle.

Key Concepts

Midpoint Properties

Understanding midpoint properties is essential when solving geometry problems involving polygons and triangles.

In the context of our exercise, a midpoint of a segment is the point that divides the segment into two segments of equal length. For instance, if M is the midpoint of segment BO in square ABCD, then BM and MO are congruent; in mathematical terms, we write this as \( BM = MO \) Similarly, with N being the midpoint of CD, CN and ND are equal in length, \( CN = ND \) It is these properties of a midpoint that are used as a foundation to prove the specific characteristics of triangle AMN in our geometry problem.

Properties of Squares

The properties of squares are numerous and fundamental to understanding many geometric concepts. A square is a regular polygon with four equal sides and four right angles.

Key properties include:

• Each interior angle is a right angle, \(90^\circ\).
• Opposite sides are parallel.
• All sides are congruent to each other.
• The diagonals of the square are equal in length and bisect each other at right angles.
• The diagonals also bisect the square's angles.

These properties help deduce further information about the geometric items derived from squares, such as the triangle AMN in this exercise.

Pythagorean Theorem

The Pythagorean theorem is a staple in mathematics, particularly in the understanding of right-angled triangles. It is formulated as \( a^2 + b^2 = c^2 \) where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides.

In our square ABCD, applying the Pythagorean theorem to the construction of triangle AMN allows us to state that if AMN is a right-angled triangle, then the lengths of AM, AN, and MN must adhere to the Pythagorean relation. This theorem was used to show that side AM is equal to side AN, since both sides consist of the square's side and half of its diagonal, thus providing crucial evidence that triangle AMN must be isosceles.

Interior Angles of a Square

An interior angle is an angle formed by two sides of a polygon on the inside. In the case of a square, there are a few important points to know:

The sum of the interior angles of any square is always \(360^\circ\), with each angle measuring \(90^\circ\) since a square is a special case of a rectangle. In our problem, by connecting the midpoints M and N to the center O, we created two right angles at O by bisecting the square's right angle, demonstrating how the square's interior angles help ascertain angle MON as a right angle. This helped establish that triangle AMN is indeed right-angled.