45. Let XYZ be an equilateral triangle and let P, Q, R be the midpoints of YZ, XZ, and XY, respectively. If:

r = Area(△PQR) / Area(△XYZ)

then find the value of r.

Find the Area Ratio of Triangle PQR and Equilateral Triangle XYZ

Understanding the Given Geometry Problem

This question is based on the relationship between a triangle and the smaller triangle formed by joining the midpoints of its three sides. The original triangle XYZ is equilateral, and the points P, Q, and R are the midpoints of its three sides.

The point P is the midpoint of YZ, the point Q is the midpoint of XZ, and the point R is the midpoint of XY. When these three midpoint points are joined, they form the inner triangle PQR.

We need to calculate the ratio:

r = Area(△PQR) / Area(△XYZ)

The central idea is that the sides of triangle PQR are half the corresponding sides of triangle XYZ. Since the area of similar triangles changes as the square of the ratio of their corresponding sides, the required ratio can be found directly.

Let the Side of Equilateral Triangle XYZ Be a

Suppose the side length of the equilateral triangle XYZ is:

XY = YZ = ZX = a

Since P is the midpoint of YZ:

YP = PZ = a/2

Since Q is the midpoint of XZ:

XQ = QZ = a/2

Similarly, since R is the midpoint of XY:

XR = RY = a/2

Thus, each side of the original equilateral triangle is divided into two equal parts.

Using the Midpoint Theorem to Find the Sides of Triangle PQR

The midpoint theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is equal to half its length.

Finding the Length of PQ

The points P and Q are the midpoints of the sides YZ and XZ, respectively. Therefore, by the midpoint theorem:

PQ ∥ XY

and:

PQ = XY/2 = a/2

Finding the Length of QR

The points Q and R are the midpoints of the sides XZ and XY, respectively. Therefore:

QR ∥ YZ

and:

QR = YZ/2 = a/2

Finding the Length of RP

The points R and P are the midpoints of the sides XY and YZ, respectively. Therefore:

RP ∥ XZ

and:

RP = XZ/2 = a/2

Hence:

PQ = QR = RP = a/2

Therefore, triangle PQR is also an equilateral triangle.

Comparing Triangle PQR With Triangle XYZ

The side length of the original triangle XYZ is a, whereas the side length of the smaller triangle PQR is a/2.

Therefore, the ratio of their corresponding sides is:

PQ/XY = (a/2)/a

Thus:

PQ/XY = 1/2

Since both triangles are equilateral, they are similar. Therefore:

△PQR ∼ △XYZ

The ratio of the corresponding sides of these two similar triangles is:

1 : 2

Using the Area Ratio of Similar Triangles

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Therefore:

Area(△PQR) / Area(△XYZ) = (PQ/XY)²

We have already found that:

PQ/XY = 1/2

Hence:

Area(△PQR) / Area(△XYZ) = (1/2)²

Therefore:

Area(△PQR) / Area(△XYZ) = 1/4

Since the question defines:

r = Area(△PQR) / Area(△XYZ)

we obtain:

r = 1/4

Alternative Solution Using the Area Formula for an Equilateral Triangle

The result can also be verified using the standard formula for the area of an equilateral triangle. If the side length of an equilateral triangle is s, its area is:

Area = (√3/4)s²

For the original equilateral triangle XYZ, the side length is a. Therefore:

Area(△XYZ) = (√3/4)a²

The side length of the inner equilateral triangle PQR is a/2. Therefore:

Area(△PQR) = (√3/4)(a/2)²

Simplifying:

Area(△PQR) = (√3/4)(a²/4)

Therefore:

Area(△PQR) = √3a²/16

Now, the required ratio is:

r = Area(△PQR) / Area(△XYZ)

Substituting the two areas:

r = (√3a²/16) / (√3a²/4)

Cancelling the common factors √3 and a²:

r = (1/16)/(1/4)

Therefore:

r = 1/4

Geometrical Explanation of the Area Ratio

Joining the midpoints of the three sides of triangle XYZ divides the original equilateral triangle into four smaller congruent equilateral triangles. These are the three corner triangles and the central triangle PQR.

Since all four smaller triangles have the same side length a/2, they have equal areas. Therefore, triangle PQR occupies exactly one of the four equal parts of the original triangle.

Hence:

Area(△PQR) = 1/4 × Area(△XYZ)

Therefore:

r = 1/4

Final Answer

The ratio of the area of the midpoint triangle PQR to the area of the original equilateral triangle XYZ is:

r = Area(△PQR) / Area(△XYZ)

r = (1/2)²

Therefore:

r = 1/4

Correct Answer: 1/4