Geometry plays a very important role when it comes to the preparation of competitive examinations. The questions on geometry are asked in almost every competitive examination like CAT, XAT, NMAT, SNAP, etc. One important area of the geometry is similar triangles. In this article you are going to learn about the basics of similar triangles and its applications in some advanced questions.
What are similar triangles?
Similar Triangles properties
1. Tests for similarity
Following are the similarity tests which can be used to check the similarity of the triangles.
Now we will discuss an example which can be solved by using the similarity of the triangles.
Example 1: In ∠ ABC, ∠ B = 90°. □ PQRS is a square of side 5 units and □ WJYZ is a square of side 10 units. MS and SY are the length and breadth respectively of rectangle MNSY. What is the length of MS?
Solution: Here ∆ ABC is a right angled triangle. ∠ ABC = 90°.
Let ∠ BAC = Θ and ∠ BCA = 90° - Θ. As PQRS and WJYZ are squares, so ∠ MPQ = Θ and ∠ PMQ = 90° - Θ. Again ∠ NJW = 90° - Θ and ∠ WNJ = Θ. Let MS = NY = x. So MQ = x - 5 and NW = x - 10. In triangles ∆MQP and ∆NWJ, ∠ PMQ = ∠ NJW, ∠ MPQ = ∠ WNJ and ∠ MQP = ∠ NWJ. Hence the angles of the two triangles are equal. So ∆MQP and ∆NWJ are similar triangles and in the similar triangles the sides opposite to equal angles are in proportion. Hence we have => (x - 5) (x - 10) = 50 => x2 - 15x + 50 = 50 => x2 - 15x = 0 => x (x - 15) = 0 => x = 0 or 15. As x is a length, so x =MS = 15 unitsSimilar triangles: Key Learning
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