A triangle ABC has been divided into four smaller triangles P, Q, R, S whose perimeters are 16 cm, 12 cm, 4 cm and 12 cm respectively. P, R and S contain the vertices A, B and C respectively. What is the perimeter of the triangle ABC ?

Question

Prepp Team · Accepted Answer

Understanding the Divided Triangle Problem

The question describes a triangle ABC that has been divided into four smaller triangles, named P, Q, R, and S. We are given the perimeters of these four smaller triangles: 16 cm, 12 cm, 4 cm, and 12 cm. We are also told that triangle P contains vertex A, triangle R contains vertex B, and triangle S contains vertex C. Triangle Q is the remaining fourth triangle.

Our goal is to find the perimeter of the original triangle ABC.

Analyzing the Triangle Division Structure

The description "P, R and S contain the vertices A, B and C respectively" suggests that triangles P, R, and S are located at the corners of the original triangle ABC. Triangle Q, being the fourth triangle, is likely in the central region.

A common way to divide a triangle into four smaller triangles with three at the corners and one in the center is by connecting the midpoints of the sides. In this case, the four resulting triangles are congruent (if the original triangle is equilateral) or similar to the original triangle, and crucially, they all have the same perimeter, which is exactly half the perimeter of the original triangle.

Let's check if this midsegment division fits the given information. If the perimeters were equal, say \(x\), then the perimeter of ABC would be \(2x\). The given perimeters are 16, 12, 4, and 12 cm. Since these are not equal, the division is not by connecting the midpoints of the sides of triangle ABC.

Deriving the Relationship of Perimeters

Despite the unequal perimeters, the structure described (three corner triangles P, R, S containing vertices A, B, C and a central triangle Q) implies a specific type of division. This division is formed by drawing three line segments inside the triangle that meet at a point, and whose endpoints lie on the sides of the original triangle. For example, if points D, E, and F are taken on sides BC, AC, and AB respectively, and segments DE, EF, and FD are drawn, they form the central triangle Q (DEF), leaving the corner triangles P (AFE), R (BDF), and S (CDE). In this configuration, P contains A, R contains B, and S contains C.

Let the sides of triangle ABC be \(a, b, c\).

The perimeter of triangle ABC is \(P_{ABC} = a + b + c\).

Let the perimeters of the four smaller triangles be \(P_P, P_Q, P_R, P_S\).

The sum of the perimeters of the four small triangles is \(P_P + P_Q + P_R + P_S\). Let's consider the segments that make up these perimeters.

Segments that lie along the sides of the original triangle ABC (segments of AB, BC, CA) are counted once in the sum of perimeters of the small triangles.
Segments that are internal lines dividing the triangle are counted twice in the sum of perimeters of the small triangles, because each internal segment is a side for two adjacent smaller triangles.

So, the sum of the perimeters of the small triangles is equal to the perimeter of the large triangle plus twice the sum of the lengths of the internal segments:

\[ P_P + P_Q + P_R + P_S = P_{ABC} + 2 \times (\text{Sum of lengths of internal segments}) \]

In the specific configuration described (P, R, S at vertices, Q central, formed by segments DE, EF, FD where D on BC, E on AC, F on AB), the internal segments are exactly the sides of the central triangle Q: DE, EF, and FD. Therefore, the sum of the lengths of the internal segments is equal to the perimeter of triangle Q.

\[ \text{Sum of lengths of internal segments} = DE + EF + FD = P_Q \]

Substituting this into the equation above:

\[ P_P + P_Q + P_R + P_S = P_{ABC} + 2 \times P_Q \]

Rearranging the equation to find the perimeter of triangle ABC:

\[ P_{ABC} = P_P + P_R + P_S + P_Q - 2 \times P_Q \] \[ P_{ABC} = P_P + P_R + P_S - P_Q \]

This formula relates the perimeter of the original triangle to the perimeters of the four smaller triangles in this specific division configuration.

Calculating Triangle ABC Perimeter

We are given the following perimeters for the smaller triangles:

Perimeter of P (\(P_P\)) = 16 cm
Perimeter of Q (\(P_Q\)) = 12 cm
Perimeter of R (\(P_R\)) = 4 cm
Perimeter of S (\(P_S\)) = 12 cm

Now, we can use the formula we derived:

\[ P_{ABC} = P_P + P_R + P_S - P_Q \]

Substitute the given values:

\[ P_{ABC} = 16 \, \text{cm} + 4 \, \text{cm} + 12 \, \text{cm} - 12 \, \text{cm} \] \[ P_{ABC} = (16 + 4 + 12) \, \text{cm} - 12 \, \text{cm} \] \[ P_{ABC} = 32 \, \text{cm} - 12 \, \text{cm} \] \[ P_{ABC} = 20 \, \text{cm} \]

Thus, the perimeter of triangle ABC is 20 cm.

Triangle	Contains Vertex	Perimeter (cm)
P	A	16
Q	(Central)	12
R	B	4
S	C	12

Using the formula: Perimeter(ABC) = \(P_P + P_R + P_S - P_Q\)

Perimeter(ABC) = 16 + 4 + 12 - 12 = 20 cm.

Revision Table: Triangle Perimeter Calculation

Concept	Description	Formula Used
Perimeter of a triangle	The total length of its boundary (sum of side lengths).	\(a+b+c\)
Sum of Perimeters (Divided Triangle)	Sum of perimeters of smaller triangles equals the main perimeter plus twice the sum of internal boundaries.	\(\sum P_{\text{small}} = P_{\text{large}} + 2 \times \sum L_{\text{internal}}\)
Specific Division Structure	Triangle ABC divided into P (at A), R (at B), S (at C), and Q (central), where internal boundaries form Q.	\(\sum L_{\text{internal}} = P_Q\)
Perimeter of ABC	Derived from the sum of perimeters and internal boundaries for this specific structure.	\(P_{ABC} = P_P + P_R + P_S - P_Q\)

Additional Information: Triangle Division Properties

When a triangle is divided into smaller regions, the relationship between the perimeters depends heavily on how the division is made. Here are some points to consider:

Midsegment Triangle: Connecting the midpoints of the sides of a triangle forms four smaller triangles. All four of these triangles are similar to the original triangle and congruent to each other. The perimeter of each small triangle is exactly half the perimeter of the original triangle.
Cevians: Line segments from a vertex to a point on the opposite side are called cevians. If three cevians intersect at a single point (like medians, angle bisectors, altitudes if concurrent), they divide the triangle into six smaller triangles. The perimeters and areas of these triangles relate in complex ways depending on the specific cevians.
General Internal Division: For a general division of a polygon into smaller polygons, the sum of the perimeters of the smaller polygons is equal to the perimeter of the original polygon plus twice the sum of the lengths of the internal dividing lines. This is the principle used to derive the formula in this problem. The specific configuration in this problem allowed the sum of internal lengths to be related directly to the perimeter of Q.

This problem uses a specific division where the internal segments form the boundaries of the central triangle, leading to a direct relationship between the perimeters.