Question

In the figure given below (not to scale), \(\mathrm{AM}: \mathrm{MC}=3: 4, \mathrm{BP}: \mathrm{PM}=3: 2\) and \(\mathrm{BN}=12 \mathrm{~cm}\). Find AN. (1) \(10 \mathrm{~cm}\) (2) \(12 \mathrm{~cm}\) (3) \(14 \mathrm{~cm}\) (4) \(16 \mathrm{~cm}\)

Short answer

Question: In triangle ABC, point M is the midpoint of BC, point N is the midpoint of CA, AM:MC = 3:4 and BP:PM = 3:2. If BN = 12 cm, find the length of AN. Answer: 8 cm.

Step-by-step solution

Set up the relationship between the segments according to given ratios

As AM:MC = 3:4 and BP:PM = 3:2, Let AM = 3x, MC = 4x, BP = 3y, and PM = 2y.

Use the segment addition postulate to find the length of BN in terms of x and y

According to segment addition postulate, AN = AM + MN, and BN = BM + MN, where MN is the length of the segment connecting points M and N. As BM = BP + PM, we have BM = 3y + 2y, which gives BM = 5y. So, BN = 5y + MN.

Find the value of MN in terms of x and y

Since M is the midpoint of BC, CM = (ΔABC's base)/2. Now, ΔABC's base = AC, which can be written as AC = AN + NC or AN + MC - AM = 7x. As CM = (ΔABC's base) / 2, we have 4x = 7x / 2. Now, x = 4 * 2 / 7, so x = 8 / 7. As AN = AM + MN, MN = AN - AM = 7x - 3x = 4x.

Substitute the value of x into MN, and find MN in terms of y

From the above step, MN = 4 * 8 / 7 = 32 / 7. Now, as MN = 32 / 7 and BN = 5y + MN, we have 12 = 5y + 32 / 7.

Solve for y

From the previous step, we have the equation 12 = 5y + 32 / 7. Now, let's rearrange the equation to find the value of y. 12 - 32 / 7 = 5y -> (84 - 32) / 7 = 5y -> 52 / 7 = 5y Solving for y, we get y = (52 / 7) / 5 -> y = 52 / 35.

Calculate the length of AN

As AN = AM + MN, we have AN = 3x + 32 / 7 and x = 8 / 7. Now, AN = 3 * (8 / 7) + 32 / 7 -> AN = (24 + 32) / 7 -> AN = 56 / 7. So, the length of AN is 8 cm. Hence, the correct option is (1) 8 cm.

Key concepts

Segment Ratio Computation

The segment ratio computation allows us to quantitatively compare different parts of a line segment using a simple mathematical relationship known as a ratio. In practical terms, if a line segment AB is divided into two parts by a point C, and we are given the ratio in which C divides AB such as AC:CB equals to some ratio like 3:2, this ratio gives us a relative measure of the lengths AC and CB.

Using variables to represent unknown segments, such as in our example where AM is represented by 3x and MC by 4x, is a common strategy for solving problems involving segment ratios. This allows the problem to be approached algebraically, transforming geometric relationships into equations that can be solved to find the measure of unknown segments.

Line Segment Division in Ratios

Line segment division in ratios involves cutting a line segment into parts that are in a specific proportional relationship. When we say that a point M divides a segment AB in the ratio of 3:2, for instance, we mean that the ratio of the length of segment AM to the length of segment MB is 3 to 2.

This concept can be visualized by considering a stick being cut into parts - each part representing a fraction of the whole stick, with the fractions being determined by the given ratio. In our geometric problems, once we assign variables to represent these unknown fractions of the stick (such as 3x for AM and 2x for MB), we can use these representations to create and solve an equation that will provide us with the concrete measures of each segment.

Mathematical Problem-Solving

Mathematical problem-solving in geometry often relies on a systematic approach to find unknown quantities. A step-by-step method, as in the segment addition problem, eases the complexity by breaking it down into smaller and more manageable parts.

Each step typically involves setting up relationships, applying postulates or theorems, performing algebraic manipulations, and checking the solution for accuracy. For instance, we used the segment addition postulate to express AN as the sum of AM and MN and created equations based on the given ratios to express lengths in terms of the variable x and then solved for x. A logical sequence of steps is crucial to ensure mathematical solutions are coherent and to guide students through the problem-solving process in a structured and understandable manner.

Geometry Principles for Line Segments

In geometry, principles for line segments provide foundational rules used to understand and solve problems related to linear structures. The segment addition postulate, for instance, states that if point B lies between points A and C on a line segment AC, then the sum of the lengths of segments AB and BC is equal to the length of AC.

The principles also cover relationships between segments created by ratios, such as our scenario where BP:PM is 3:2, indicating a specific proportional division of a line segment. Understanding these principles is essential for solving geometrical problems, as they form the basis for reasoning, construction of equations, and direct proofs. Correct application ensures accurate and dependable solutions to common geometric questions.