Triangles — Karnataka (SSLC) Class 10 Mathematics Solutions (Free)

Basic Proportionality Theorem: if a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, then it divides those two sides in the same ratio. In triangle ABC, if a line parallel to BC meets AB at D and AC at E, then: AD/DB = AE/EC What it tells us about similar triangles: this theorem is the foundation for proving that triangle ADE is similar to triangle ABC. Since DE is parallel to BC, corresponding angles are equal (∠ADE = ∠ABC and ∠AED = ∠ACB, both by the corresponding angles formed with a transversal), and combined with the shared angle at A, this gives AA similarity between the two triangles — meaning triangle ADE is simply a smaller, proportionally scaled copy of triangle ABC. This is exactly why the ratio AD/DB equals AE/EC: both ratios describe the same underlying scale factor between the two similar triangles.

Since the triangles are similar, their corresponding sides are in the same ratio (the scale factor): Scale factor = PQ/AB = 10/5 = 2 Since the triangles are similar, EVERY pair of corresponding sides shares this same scale factor: QR = BC × scale factor = 8 × 2 = 16 cm RP = CA × scale factor = 7 × 2 = 14 cm QR = 16 cm, RP = 14 cm.

By the Basic Proportionality Theorem, since DE ∥ BC: AD/DB = AE/EC Given: AD = 4 cm, DB = 8 cm, AE = 5 cm. Find EC. 4/8 = 5/EC Cross-multiplying: 4 × EC = 8 × 5 4 × EC = 40 EC = 10 cm

For similar triangles, the ratio of their AREAS equals the SQUARE of the ratio of their corresponding sides — not the same ratio as the sides themselves. Given: side ratio = 2:7, area of smaller triangle = 20 cm². Area ratio = (side ratio)² = (2/7)² = 4/49 So: Area of smaller / Area of larger = 4/49 20 / Area of larger = 4/49 Area of larger = 20 × 49/4 = 245 cm² The area of the larger triangle is 245 cm². (This squared relationship is why doubling a triangle's side lengths doesn't just double its area — it quadruples it.)

Since DE ∥ BC, triangle ADE is similar to triangle ABC (by AA similarity, as explained by the Basic Proportionality Theorem). Given AD:DB = 3:2, the whole side AB is divided into 3 + 2 = 5 equal parts, with AD representing 3 of those 5 parts. So AD/AB = 3/5 Since the triangles are similar, corresponding sides share the same ratio: DE/BC = AD/AB DE/20 = 3/5 DE = 20 × 3/5 = 12 cm

Given: triangle ABC, right-angled at B, with BD drawn perpendicular to hypotenuse AC, meeting it at D. To prove: AB² = AD × AC Proof: Step 1: In triangles ABD and ABC: ∠ADB = ∠ABC = 90° (BD ⊥ AC by construction, and ABC is given as the right angle) ∠A is common to both triangles. Step 2: By AA similarity, △ABD ~ △ACB (note the order of vertices, matching corresponding angles). Step 3: Since the triangles are similar, their corresponding sides are proportional: AB/AC = AD/AB Step 4: Cross-multiplying: AB × AB = AD × AC AB² = AD × AC Hence proved. (This result, along with its counterpart BC² = CD × CA, together give a geometric proof of the Pythagorean theorem when added: AB² + BC² = AD×AC + CD×AC = AC×(AD+CD) = AC×AC = AC².)

For similar triangles, the ratio of their PERIMETERS equals the same ratio as their corresponding sides (unlike area, which squares the ratio). Given: side ratio = 5:9, perimeter of smaller triangle = 40 cm. Perimeter of smaller / Perimeter of larger = 5/9 40 / Perimeter of larger = 5/9 Perimeter of larger = 40 × 9/5 = 72 cm The perimeter of the larger triangle is 72 cm.

Two triangles are similar (by the SSS similarity criterion) if the ratios of ALL THREE pairs of corresponding sides are equal. Checking each ratio: PQ/AB = 8/6 = 4/3 QR/BC = 12/9 = 4/3 RP/CA = 16/12 = 4/3 All three ratios equal 4/3. Since all corresponding sides are in the same ratio, the triangles ARE similar by the SSS (Side-Side-Side) similarity criterion, with a scale factor of 4/3 (triangle PQR is a proportionally larger version of triangle ABC).

Since DE ∥ BC, by the Basic Proportionality Theorem: AD/AB = AE/AC Given: AD = 3 cm, AB = 8 cm, AC = 16 cm. Find AE. 3/8 = AE/16 AE = 16 × 3/8 = 6 cm Now find EC: EC = AC − AE = 16 − 6 = 10 cm AE = 6 cm, EC = 10 cm.