In triangle ABC, P and Q are the mid points of AB and AC, respectively. R is a point on PQ such that PR : RQ = 3 : 5 and QR = 20 cm, then what is the length (in cm) of BC?

Question

Prepp Team · Accepted Answer

Solving the Triangle Midpoint Problem

The problem involves a triangle ABC where P and Q are the midpoints of sides AB and AC, respectively. We are given information about a point R on the segment PQ and need to find the length of the base BC.

Understanding the Midpoint Theorem

The Midpoint Theorem is crucial for solving this problem. It states that the line segment connecting the midpoints of two sides of a triangle is:

Parallel to the third side.
Exactly half the length of the third side.

In triangle ABC, since P is the midpoint of AB and Q is the midpoint of AC, the segment PQ is parallel to BC and its length is half the length of BC. Mathematically, this means:

\( \text{PQ} \parallel \text{BC} \)

\( \text{PQ} = \frac{1}{2} \times \text{BC} \)

Therefore, if we can find the length of PQ, we can find the length of BC by doubling PQ: \( \text{BC} = 2 \times \text{PQ} \).

Using the Given Ratio to Find PQ

We are given that R is a point on the segment PQ such that the ratio of PR to RQ is 3 : 5. We are also given that the length of RQ is 20 cm.

The ratio PR : RQ = 3 : 5 can be written as:

\( \frac{\text{PR}}{\text{RQ}} = \frac{3}{5} \)

We know RQ = 20 cm. Substituting this value:

\( \frac{\text{PR}}{20} = \frac{3}{5} \)

Now, we can solve for the length of PR:

\( \text{PR} = \frac{3}{5} \times 20 \)

\( \text{PR} = 3 \times \frac{20}{5} \)

\( \text{PR} = 3 \times 4 \)

\( \text{PR} = 12 \text{ cm} \)

The total length of the segment PQ is the sum of PR and RQ:

\( \text{PQ} = \text{PR} + \text{RQ} \)

\( \text{PQ} = 12 \text{ cm} + 20 \text{ cm} \)

\( \text{PQ} = 32 \text{ cm} \)

Calculating the Length of BC

Now that we have the length of PQ (32 cm), we can use the Midpoint Theorem relationship \( \text{BC} = 2 \times \text{PQ} \) to find the length of BC.

\( \text{BC} = 2 \times 32 \text{ cm} \)

\( \text{BC} = 64 \text{ cm} \)

Therefore, the length of BC is 64 cm.

Step	Description	Calculation
1	Use the ratio PR : RQ to find PR.	\( \frac{\text{PR}}{20} = \frac{3}{5} \implies \text{PR} = 12 \text{ cm} \)
2	Calculate the length of PQ.	\( \text{PQ} = \text{PR} + \text{RQ} = 12 + 20 = 32 \text{ cm} \)
3	Use the Midpoint Theorem to find BC.	\( \text{BC} = 2 \times \text{PQ} = 2 \times 32 = 64 \text{ cm} \)

Revision Table: Key Concepts

Concept	Description	Relevance to Problem
Midpoint	A point that divides a line segment into two equal parts. P is the midpoint of AB, Q is the midpoint of AC.	Defines the segment PQ.
Ratio	A comparison of two quantities (PR and RQ) using division.	Used to determine the lengths of PR and PQ.
Midpoint Theorem	Segment joining midpoints of two sides is parallel to and half the third side.	Connects the length of PQ to the length of BC.

Additional Information on Midpoint Theorem and Applications

The Midpoint Theorem is a powerful tool in geometry, especially when dealing with triangles and parallel lines. Here are some additional points:

Converse of Midpoint Theorem: A line drawn through the midpoint of one side of a triangle, parallel to another side, bisects the third side. This is the reverse of the main theorem.
Varignon's Theorem: The midpoints of the sides of any quadrilateral form a parallelogram. This theorem can be proven using the Midpoint Theorem repeatedly on the triangles formed by the quadrilateral's diagonals.
Applications: The Midpoint Theorem is used in various geometric proofs and constructions. It helps establish relationships between line segments within a triangle. It's a fundamental concept in coordinate geometry as well.
Vector Approach: The Midpoint Theorem can also be proven using vectors, which offers a different perspective on this geometric property.