Tiêu đề 6. TRIANGLES.pdf ✅

CLASS - X MATHEMATICS 113 FOUNDATION 6. TRIANGLES VOL - I Introduction Similar Figures Similarity of Triangles Theorems on Similarity Criterion Basic Proportionality Theorem and its converse Baudhayan / Pythagoras Theorem and its converse A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted triangle ∆ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it. Looking around you will see many objects which are of the same shape but of same or different sizes. For examples, leaves of a tree have almost the same shape but same or different sizes. Similarly, photographs of different sizes developed from the same negative are of same shape but different sizes, the miniature model of a building and the building itself are of same shape but different sizes. All those objects which have the same shape but not necessarily the same size are called similar objects SYNOPSIS-1 1.1 SIMILAR FIGURES Two figures are said to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are congruent, and the ratios of the lengths of their corresponding sides are equal. This common ratio is called the scale factor. The symbol ~ is used to indicate similarity. For example: 1. In the figure given below ABCDE ~ pentagon VWXYZ. CHAPTER 6 TRIANGLES
114 MATHEMATICS VOL - I CLASS - X 6. TRIANGLES FOUNDATION 2. In the given figure, the hexagon A1 B1 C1 D1 E1 F1 is flipped horizontally to get A2 B2 C2 D2 E2 F2 . Then hexagon A2 B2 C2 D2 E2 F2 is translated to get A3 B3 C3 D3 E3 F3 . Hexagon A3 B3 C3 D3 E3 F3 is dilated by a scale factor of 1 2 to get A4 B4 C4 D4 E4 F4 . Note that A1 B1 C1 D1 E1 F1 ~ A2 B2 C2 D2 E2 F2 ~ A3 B3 C3 D3 E3 F3 ~ A4 B4 C4 D4 E4 F4 . That is, all four hexagons are similar. (In fact, the first three are congruent) 1.2 SIMILAR TRIANGLES Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles look the same but the sizes can be different. In general, similar triangles are different from congruent triangles. There are various methods by which we can find if two triangles are similar or not. Let us learn more about similar triangles and their properties. Two triangles will be similar if the angles are equal (corresponding angles) and sides are in the same ratio or proportion (corresponding sides). Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides must be the same. If two triangles are similar that means, i) All corresponding angle pairs of triangles are equal. ii) All corresponding sides of triangles are proportional. We use the “~” symbol to represent the similarity. So, if two triangles are similar, we show it as ∆QPR ~ ∆XYZ 1.3 AREAS OF SIMILAR TRIANGLES The ratio of the area of two similar triangles has relation with the ratio of the corresponding sides. The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Statement: The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Given: ∆ABC ∼ ∆DEF. so, AB DE BC EF AC DF = = Also, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F To prove: area ABC area DEF AB DE BC EF AC DF ∆ ∆ = = = 2 2 2 2 2 2 Construction: Through A draw AP ⊥ BC and through D draw DQ ⊥ EF. Proof: area (∆ABC) = × 1 2 BC AP and area (∆DEF) = × 1 2 EF DQ
CLASS - X MATHEMATICS 115 FOUNDATION 6. TRIANGLES VOL - I Thus, area ABC area DEF BC AP EF DQ area ABC area DEF BC EF AP DQ ∆ ∆ ∆ ∆ = × × ⇒ = × 1 2 1 2 ....(i) In ∆APB and ∆DQE, ∠1 = ∠2 = 90° [By construction] ∠3 = ∠4 [Given] ∴ ∆APB ∼ ∆DQE [AA corollary] ∴ AB DE AP DQ BP EQ AB DE AP DQ = = ⇒ = ...(ii) Also, AB DE BC EF = [Given]....(iii) From (ii) and (iii), we get AP DQ BC EF = ...(iv) Putting the value of AP DQ from (iv) in (i), we get area ABC area DEF BC EF BC EF BC EF ∆ ∆ = × = 2 2 ....(v) Similarly, it can also be proved that area ABC area DEF AB DE ∆ ∆ = 2 2 ...(vi) and area ABC area DEF AC DF ∆ ∆ = 2 2 ...(vii) From (v), (vi) and (vii), we obtain area ABC area DEF AB DE BC EF AC DF ∆ ∆ = = = 2 2 2 2 2 2 1.4 SIMILARITY OF TRIANGLES Similar triangles are triangles for which the corresponding angle pairs are equal. That means equiangular triangles are similar. Therefore, all equilateral triangles are examples of similar triangles. The following image shows similar triangles, but we must notice that their sizes are different. ∆QPR ~ ∆XYZ We can find out or prove whether two triangles are similar or not using the similarity theorems. We use these similarity criteria when we do not have the measure of all the sides of the triangle or measure of all the angles of the triangle. These similar triangle theorems help us quickly find out whether two triangles are similar or not. There are three major types of similarity rules, as given below, AA (or AAA) or Angle-Angle Similarity Theorem SAS or Side-Angle-Side Similarity Theorem SSS or Side-Side-Side Similarity Theorem Same shape, but not necessarily the same size.
116 MATHEMATICS VOL - I CLASS - X 6. TRIANGLES FOUNDATION Corresponding angles are equal. Corresponding sides are in the same ratio. a p b q c r = = To test for similar triangles: AA - If 2 corresponding angles are equal. SSS - If 3 corresponding sides are in the same ratio. SAS - Ratio of 2 pairs of corresponding sides equal and their included angles are equal. Let us understand these similar triangles theorems with their proofs. THEOREM 1: AA (OR AAA) OR ANGLE-ANGLE SIMILARITY CRITERION AA similarity criterion states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles. AA similarity rule is easily applied when we only know the measure of the angles and have no idea about the length of the sides of the triangle. In the image given below, if it is known that ∠B = ∠G, and ∠C = ∠F. And we can say that by the AA similarity criterion, ∆ABC and ∆EGF are similar or ∆ABC ~∆EGF. AB EG BC GF AC EF = = and ∠A = ∠E. THEOREM 2: SAS OR SIDE-ANGLE-SIDE SIMILARITY CRITERION According to the SAS similarity theorem, if any two sides of the first triangle are in exact proportion to the two sides of the second triangle along with the angle formed by these two sides of the individual triangles are equal, then they must be similar triangles. This rule is generally applied when we only know the measure of two sides and the angle formed between those two sides in both the triangles respectively. In the image given below, if it is known that AB DE AC DF = , and ∠A = ∠D And we can say that by the SAS similarity criterion, ∆ABC and ∆DEF are similar or ∆ABC ~ ∆DEF. THEOREM 3: SSS OR SIDE-SIDE-SIDE SIMILARITY CRITERION According to the SSS similarity theorem, two triangles will similar to each other if the corresponding ratio of all the sides of the two triangles are equal. This criterion is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.