Question

In trapezium KLMN, KL and MN are parallel sides. A line is drawn, from the point \(A\) on KN, parallel to MN meeting LM at B. KN : LM is equal to (1) KL: NM (2) \((\mathrm{KL}+\mathrm{KA}):(\mathrm{NM}+\mathrm{BM})\) (3) \((\mathrm{KA}-\mathrm{AN}):(\mathrm{LB}-\mathrm{BM})\) (4) \(\mathrm{KL}^{2}: \mathrm{MN}^{2}\)

Short answer

Answer: The correct ratio of the sides KN to LM is \((\mathrm{KL}+\mathrm{KA}):(\mathrm{NM}+\mathrm{BM})\).

Step-by-step solution

Analyze Trapezium KLMN

In trapezium KLMN, sides KL and MN are parallel. We are also given that line segment AB is parallel to MN. Hence, by the transversal properties, angle KAN = angle ALM and angle NAM = angle MNL.

Apply the AA Similarity criterion

We can see that triangles KAN and LAM have the following pairs of equal angles: KAN = ALM and NAM = MNL. Thus, by the AA Similarity criterion, we can conclude that triangles KAN and LAM are similar.

Write the side ratios of similar triangles

Since triangles KAN and LAM are similar, we can write the ratios of their corresponding sides: \(\frac{\mathrm{KA}}{\mathrm{AL}} = \frac{\mathrm{AN}}{\mathrm{LM}}\)

Find the ratio KN : LM

In triangle KAN, KN = KA + AN. Adding AN to both the sides of the above ratio, we get \(\frac{\mathrm{KA}+\mathrm{AN}}{\mathrm{AL}} = \frac{\mathrm{KN}}{\mathrm{LM}}\). Thus, the correct option is: (2) \((\mathrm{KL}+\mathrm{KA}):(\mathrm{NM}+\mathrm{BM})\)

Key concepts

Similar Triangles

Similar triangles are two triangles that have the same shape but may differ in size. The main characteristic of similar triangles is that their corresponding angles are equal and their corresponding sides are in proportion. This property is useful in geometry to find missing lengths and angles. In the case of trapezium KLMN, the triangles KAN and LAM are similar. This is because the angles are equal due to parallel lines.
When two triangles are similar, you can use the similarity to set up a ratio of their corresponding sides. This helps in solving geometric problems that require finding unknown lengths.

Parallel Lines

Parallel lines are lines in a plane that are equidistant from each other and never intersect. In a trapezium, one pair of opposite sides is parallel. In trapezium KLMN, the sides KL and MN are parallel. By drawing line segment AB parallel to MN, we ensure that certain angles in the trapezium and triangles formed are equal.
This property of parallel lines allows us to form similar triangles, as we see with triangles KAN and LAM. The role of parallel lines is crucial because they facilitate the angle relationships necessary for similarity.

AA Similarity Criterion

The AA (Angle-Angle) similarity criterion is a rule that determines when two triangles are similar. If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. This means that their corresponding sides are in proportion.
For trapezium KLMN, triangles KAN and LAM are shown to be similar through the AA similarity criterion. This is because angle KAN equals angle ALM and angle NAM equals angle MNL, thanks to the parallel lines. When angles are confirmed to be equal, similarity of triangles can be declared, which is a powerful tool for solving for unknowns.

Side Ratios

In similar triangles, the sides that correspond are proportional. This means if you have two similar triangles, the ratio of one pair of corresponding sides will be the same as the ratio of another pair.
For the problem with trapezium KLMN, once we establish that triangles KAN and LAM are similar, we can write the side ratios. For example, \( \frac{\mathrm{KA}}{\mathrm{AL}} = \frac{\mathrm{AN}}{\mathrm{LM}} \). From these ratios, information about one side can help determine the length or proportion of another side. By manipulating these ratios, the specified problem's solution was achieved by expressing KN:LM in terms of known lengths.