Question

In \(\Delta \mathrm{PQR}, \mathrm{PQ}=6 \mathrm{~cm}, \mathrm{PR}=9 \mathrm{~cm}\) and \(\mathrm{M}\) is a point on \(\mathrm{QR}\) such that it divides \(\mathrm{QR}\) in the ratio \(1: 2 .\) \(\mathrm{PM} \perp \mathrm{QR}\). Find \(\mathrm{QR}\). (1) \(\sqrt{18} \mathrm{~cm}\) (2) \(3 \sqrt{12} \mathrm{~cm}\) (3) \(3 \sqrt{15} \mathrm{~cm}\) (4) \(\sqrt{20} \mathrm{~cm}\)

Short answer

Answer: The length of side QR in triangle PQR is 3√15 cm.

Step-by-step solution

Identify the triangles

Observe that we now have two triangles, PPM and PMQ. PM is perpendicular to QR, indicating that each of these triangles (PPM and PMQ) is a right triangle.

Determine the lengths of PM and QM using the given ratio

We are given that point M divides side QR in the ratio 1:2. Let PM = x and MQ = 2x. Since QR = PM + QM, we have QR = x + 2x = 3x.

Apply the Pythagorean theorem to triangle PPM

In triangle PPM, we have: PP^2 + PM^2 = PQ^2 PP^2 + x^2 = 6^2 PP^2 + x^2 = 36.

Apply the Pythagorean theorem to triangle PMQ

In triangle PMQ, we have: PM^2 + MQ^2 = PR^2 x^2 + (2x)^2 = 9^2 x^2 + 4x^2 = 81 5x^2 = 81.

Solve for x

From the equation 5x^2 = 81, we can solve for x as follows: x^2 = 81/5 x = sqrt{81/5} x = 3\sqrt{5/3}.

Find the length of QR

Since QR = 3x, we can find its length by plugging in the value of x we found in step 5: QR = 3(3\sqrt{5/3}) QR = 3\sqrt{15}. So, the length of QR is 3\sqrt{15} cm, which corresponds to option (3).

Key concepts

Ratio and Proportion in Geometry

Understanding the concept of ratio and proportion in geometry is pivotal for solving a wide range of problems, including those involving segments of geometric figures. A ratio represents a comparative value between two or more quantities, showing how many times one value contains another.

Proportion, on the other hand, occurs when two ratios are equivalent. In our exercise, we're given a scenario in which a point divides a segment in a specific ratio, 1:2. This directly informs us the segment is partitioned into three equal parts, an insight that is critical for correctly applying the Pythagorean theorem later on. It's essential to grasp these concepts, as they underpin not only the splitting of segments but also scaling of shapes and the creation of similar figures within geometry.

Solving Right Triangles

Solving right triangles is a fundamental skill in geometry that involves finding unknown lengths of sides and angles. Every right triangle contains a 90-degree angle, and the sides opposite and adjacent to this right angle are important for calculations.

The Pythagorean theorem is a go-to tool for these problems, as it relates the squares of the lengths of the legs to the square of the hypotenuse. Understanding the relationships between the sides and angles, as well as mastering the Pythagorean theorem, can make solving these triangles far less intimidating. By recognizing which side is the hypotenuse and which sides are the legs, one can swiftly move ahead with problem-solving.

Application of the Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, especially when working with right-angled triangles. It states that for any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, if 'a' and 'b' are the legs and 'c' is the hypotenuse, the theorem is expressed as \(a^2 + b^2 = c^2\). This theorem is not just theoretical; its practical applications are vast, including computing distances, constructing shapes, and even in real-world situations such as navigation and construction. In our problem, the use of this theorem allows us to connect the geometric concept of ratio to actually find the lengths of sides in these triangles.

Mathematical Problem-Solving

Mathematical problem-solving is at the heart of understanding and applying mathematical concepts. It involves a series of steps that guide learners from identifying the problem to finding a solution. Recognizing the type of problem, choosing appropriate strategies, and carefully executing calculations are all integral parts of this process.

In our exercise, the problem-solving process starts with identifying right triangles and divides them according to given ratios. From there, the application of the Pythagorean theorem in a stepwise manner leads to the solution. Patience and precision are critical, as is the ability to manipulate algebraic expressions when solving for unknowns. Understanding these procedural steps enhances one's ability to tackle a broad array of mathematical challenges.