then the value of r is ______.

Find the Area Ratio of the Midpoint Triangle PQR to the Equilateral Triangle XYZ

Understanding the Given Geometry Problem

This question asks us to compare the area of a smaller triangle PQR with the area of the original equilateral triangle XYZ. The points P, Q, and R are not arbitrary points. Each one is the midpoint of one side of the larger triangle.

Specifically, P is the midpoint of YZ, Q is the midpoint of XZ, and R is the midpoint of XY. When the midpoints of the three sides of any triangle are joined, the smaller triangle formed inside the original triangle is called the medial triangle or midpoint triangle.

The important geometric fact is that every side of the midpoint triangle is exactly half the length of the corresponding side of the original triangle. Therefore, △PQR and △XYZ are similar triangles with a linear scale factor of 1/2.

Since the areas of similar triangles are proportional to the squares of their corresponding side lengths, the required area ratio is:

(1/2)² = 1/4

Therefore, the value of r is 1/4.

Given Information

The original triangle XYZ is an equilateral triangle. Therefore:

XY = YZ = ZX

The points are defined as follows:

P is the midpoint of YZ

Q is the midpoint of XZ

R is the midpoint of XY

The required quantity is:

r = Area(△PQR) / Area(△XYZ)

Our objective is to calculate this ratio.

Step-by-Step Solution Using the Midpoint Theorem

Step 1: Let the Side Length of the Equilateral Triangle Be a

Let each side of the equilateral triangle XYZ have length a. Therefore:

XY = YZ = ZX = a

Since P, Q, and R are the midpoints of the respective sides, each original side is divided into two equal parts.

Therefore:

YP = PZ = a/2

XQ = QZ = a/2

XR = RY = a/2

Step 2: Apply the Midpoint Theorem to Find PQ

In triangle XYZ, P is the midpoint of YZ and Q is the midpoint of XZ. According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length.

Therefore:

PQ ∥ XY

and:

PQ = XY/2

Since XY = a:

PQ = a/2

Step 3: Find QR Using the Midpoint Theorem

The point Q is the midpoint of XZ, while R is the midpoint of XY. Therefore, QR joins the midpoints of two sides of triangle XYZ.

By the midpoint theorem:

QR ∥ YZ

and:

QR = YZ/2

Since YZ = a:

QR = a/2

Step 4: Find RP Using the Midpoint Theorem

The point R is the midpoint of XY, and P is the midpoint of YZ. Therefore, RP also joins the midpoints of two sides of triangle XYZ.

By the midpoint theorem:

RP ∥ XZ

and:

RP = XZ/2

Since XZ = a:

RP = a/2

Step 5: Identify the Shape and Size of Triangle PQR

We have found that:

PQ = QR = RP = a/2

Therefore, △PQR is also an equilateral triangle.

The side length of △XYZ is a, while the side length of △PQR is a/2. Hence, the ratio of corresponding side lengths is:

Side of △PQR / Side of △XYZ = (a/2)/a

Therefore:

Side ratio = 1/2

Step 6: Use the Area Ratio of Similar Triangles

The triangles PQR and XYZ are similar. 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) = (1/2)²

Hence:

Area(△PQR) / Area(△XYZ) = 1/4

Since:

r = Area(△PQR) / Area(△XYZ)

we obtain:

r = 1/4

Solution Using the Area Formula of an Equilateral Triangle

The result can also be verified by directly calculating the areas of both equilateral triangles.

The area of an equilateral triangle with side length s is:

Area = (√3/4)s²

Let the side length of the original equilateral triangle XYZ be a. Therefore:

Area(△XYZ) = (√3/4)a²

The side length of the midpoint 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 calculate the required ratio:

r = [√3a²/16] / [√3a²/4]

Dividing the two quantities:

r = (√3a²/16) × (4/√3a²)

The common factors √3 and a² cancel, giving:

r = 4/16

Therefore:

r = 1/4

Geometrical Interpretation of the Midpoint Triangle

Joining the midpoints P, Q, and R divides the original equilateral triangle XYZ into four smaller triangles. These are △XQR, △YPR, △ZPQ, and the central triangle △PQR.

Each smaller triangle has side length a/2. Therefore, all four smaller triangles are congruent equilateral triangles and have equal areas.

Since the complete triangle XYZ is divided into four equal-area triangles, the central triangle PQR occupies exactly one of these four equal parts.

Therefore:

Area(△PQR) = 1/4 × Area(△XYZ)

Hence:

r = 1/4

Why the Area Ratio Is Not 1/2

It may appear that because every side of △PQR is half the corresponding side of △XYZ, its area should also be half. However, area is a two-dimensional quantity, so it changes according to the square of the linear scale factor.

The side ratio is:

1/2

Therefore, the area ratio is:

(1/2)² = 1/4

This is why halving every side of a similar triangle reduces its area to one-fourth of the original area.

Alternative Solution Using Four Equal Triangles

The midpoint construction provides an especially simple visual interpretation. Because P, Q, and R are the midpoints of the sides of the equilateral triangle, each side of the original triangle is divided into two equal parts.

The three segments PQ, QR, and RP divide △XYZ into four congruent equilateral triangles:

△XQR, △YPR, △ZPQ, and △PQR

Since all four triangles are congruent, their areas are equal. Thus, each small triangle has one-fourth of the total area of △XYZ.

Therefore:

Area(△PQR) / Area(△XYZ) = 1/4

Hence:

r = 1/4

Complete Solution in Compact Form

Since P, Q, and R are the midpoints of YZ, XZ, and XY, respectively, the midpoint theorem gives:

PQ = XY/2

QR = YZ/2

RP = XZ/2

Therefore, △PQR is similar to △XYZ with corresponding side ratio:

1/2

The ratio of the areas of similar triangles is the square of the corresponding side ratio. Hence:

r = Area(△PQR) / Area(△XYZ)

= (1/2)²

= 1/4

Final Answer

The triangle PQR is the medial triangle of the equilateral triangle XYZ. Its side length is half the side length of △XYZ, so its area is one-fourth of the area of △XYZ.

Answer: r = 1/4