The Concept of Similarity
Q1–10 of 40In triangles ABC and DEF, ∠A = ∠D and ∠B = ∠E. By which criterion are the triangles similar?
Hint: You're told two pairs of angles are equal. Since the angles of any triangle add up to 180°, what does that force about the third pair of angles? Once all three angles match, the triangles are automatically similar.
Triangle ABC ~ Triangle DEF, with AB = 4, BC = 6, and DE = 8. Find EF.
Hint: Since the triangles are similar, all corresponding sides share the same scale factor. First work out the scale factor by comparing AB (in ABC) with its corresponding side DE (in DEF), then apply that same factor to BC to get EF.
Two triangles have sides in the ratio 2:3. If the area of the smaller triangle is 20 cm², find the area of the larger triangle.
Hint: Be careful — area ratio is NOT the same as side ratio. If sides are in ratio 2:3, areas are in ratio 2²:3². Use this squared ratio to scale up the given area.
In the figure, ΔABC is similar to ΔEDC.
Q. If we have AB = 4 cm, ED = 3 cm, CE = 4.2 and CD = 4.8 cm, find the value of CA and CB
Hint: Since ΔABC ~ ΔEDC, matching vertices are A↔E, B↔D, C↔C, so corresponding sides are AB/ED = BC/DC = CA/CE. Using AB/ED = 4/3: BC = (4/3)×DC = (4/3)×4.8 = 6.4 cm, and CA = (4/3)×CE = (4/3)×4.2 = 5.6 cm. So CA = 5.6 cm and CB = 6.4 cm.
The area of similar triangles, ABC and DEF are 144 cm² and 81 cm² respectively. If the longest side of larger ΔABC be 36 cm, then the longest side of smaller ΔDEF is
Hint: For similar triangles, the ratio of areas equals the square of the ratio of corresponding sides. Area ratio = 144/81, so the side ratio = √(144/81) = 12/9 = 4/3. So the longest side of DEF = 36 × (9/12) = 27 cm.
The areas of two similar Δs are respectively 9 cm² and 16 cm². Find the ratio of their corresponding sides.
Hint: Ratio of sides = √(ratio of areas) = √(9/16) = 3/4, i.e. 3 : 4.
In triangle ABC, DE || BC, with D on AB and E on AC. If AD = 3, DB = 6, and AE = 4, find EC.
Hint: This is the Basic Proportionality Theorem (BPT) — when a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. So AD/DB should equal AE/EC; just plug in the known values and solve for EC.
Triangle PQR ~ Triangle XYZ with a scale factor of 5:2. If PQ = 15 cm, find XY.
Hint: The scale factor 5:2 means every side of PQR is 5/2 times the corresponding side of XYZ. Since XY corresponds to PQ, you need to go the other way — divide PQ by (5/2), or equivalently multiply by (2/5).
A vertical pole 6 m tall casts a shadow of 4 m. At the same time, a tower casts a shadow of 28 m. Find the height of the tower.
Hint: At the same time of day, the sun's angle is the same everywhere, so the pole-and-shadow triangle is similar to the tower-and-shadow triangle. Set up height/shadow as equal ratios for both and cross-multiply.
Two isosceles Δs have equal angles and their areas are in the ratio 16 : 25. Find the ratio of their corresponding heights.
Hint: Equal angles means the triangles are similar, and for similar triangles, ratio of areas = (ratio of corresponding heights)². So height ratio = √(16/25) = 4/5