Similarity of triangles - practice problems
Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional by the same scale factor. Similarity can be proven using AA (angle-angle), SAS (side-angle-side with proportional sides), or SSS (all sides proportional) similarity theorems. Similar triangles have the same shape but not necessarily the same size. The ratio of corresponding sides is constant, and this ratio relates to areas (which scale by the square of the ratio) and volumes of similar solids (which scale by the cube). Applications include indirect measurement, map scaling, architectural models, and solving problems using shadow lengths or mirror reflections. Understanding triangle similarity is fundamental to trigonometry, coordinate geometry, and geometric proofs.Directions: Solve each problem carefully. Show all your work.
Number of problems found: 160
- Building shadow
When Sun's altitude 30° above the horizontal, then find the length of the shadow of a 50 m high of a building . - The areas
The areas of two similar triangles are 12 sq cm and 48 sq cm. If the height of the smaller one is 2.1 cm, then find the height of the bigger one. - The angles 6
If the angles of a triangle are in the ratio 2 : 3 : 4. Find the value of each angle. - Double sides
If each side of a triangle is doubled, then find the ratio of the area of the new triangle thus formed to the given triangle. - A cone 4
A cone with a radius of 10 cm is divided into two parts by drawing a plane through the midpoint of its axis parallel to its base. Compare the volumes of the two parts. - Two similar 2
Two similar polygons have corresponding sides 15 inches and 6 inches. If the area of the first is 2700 square inches, what is the area of the second? - Scale model
In a model train set, 1.38 inches represents one foot in real life. The height of One World Trade Center in New York City is 1776 feet. How tall would a scale model of the building be? Should you calculate 1776 x 1.38 or 1776 ÷ 1.38? - Similarity coefficient
Given triangle ABC with sides a = 12 cm b = 9 cm c = 7 cm and triangle DEF with sides d = 8.4 cm, e = 6.3 cm f = 4.9 cm Find out if triangles ABC and DEF are similar if so, write the similarity coefficient and according to which sentence they are similar - Trapezoid 83
Trapezoid ABCD is composed of five triangles. Points E, and G divide segment AB in the ratio 2:4:3 (in this order) into three segments. Point F is the midpoint of segment AD. Triangle AEF is isosceles and right-angled. Triangles GBC and CDG are right-angl - Shadow - an observation tower
How tall is the observation tower if it casts a shadow 9.6 m long at exactly the same moment that a half-meter pole casts a shadow 30 cm long? - Similar triangles We have similar triangles ABC with angle CAB=45° and angle ACB= 30° and a similar triangle OPN. What is the angle NOP in a similar triangle?
- Exterior angles
In triangle ABC, the size of the exterior angle at vertex C is equal to 126°. The size of the internal angles at vertices A and B are in the ratio 5: 9. Calculate the size of the internal angles α, β, γ of triangle ABC. - Circumscribed triangle Calculate the radius of the circle of the circumscribed triangle, which has side dimensions of 8, 10, and 14 cm.
- Height of poplar
From the 40 m high observation deck, you can see the top of the poplar at a depth angle of 50°10' and the bottom of the poplar at a depth angle of 58°. Calculate the height of the poplar. - Mast angles and height
Calculate the height of the mast, whose foot can be seen at a depth angle of 11° and the top at a height angle of 28°. The mast is observed from a position 10 m above the level of the base of the mast. - Building shadow height The school building casts a shadow 16 m long on the plane of the yard, and at the same time, a vertical meter pole casts a shadow 132 cm long. Determine the height of the building.
- Selection 4
Selection triangle, which is similar to the given triangle RTG. ∆ RTG, r= 24 dm, t = 28 dm, g= 30 dm. ∆ SHV= 6 dm, h= 7.5 dm, v= 7 dm ∆ VSH= v= 7 dm, s= 6 dm, h= 7.5 dm ∆ HVS= h= 7.5 dm, v= 7 dm, s = 6 dm. ∆ VHS= v= 7 dm, h = 7.5 dm, s= 6 dm. ∆ HSV= h= 7. - Triangle similarity selection
Choose a triangle that is similar to the given triangle. - ∆ TFC= t= 8 cm, f= 9 cm, c= 7 cm. : ∆ PKU= p= 45 cm, k= 35 cm, u= 40 cm. ∆ UPK= u= 40 cm, p= 45 cm, k= 35 cm. ∆ PUK= p= 45 cm, u= 40 cm, k= 35 cm. ∆ KPU= k= 35 cm, p= 45 cm, u= 40 cm. ∆ KUP= k= 35 - Triangle similarity decision Decide whether the triangles are similar. Choose between Yes/No. ∆ YUO: y= 9m, u= 17 m, o= 12 m, ∆ ZXV= z= 207 dm, x= 341 dm, v= 394 dm
- Triangle similarity area
Triangle ABC and triangle ADE are similar. Calculate in square centimeters the area of triangle ABC if the length of side DE is 12 cm, the length of side BC is 16 cm, and the area of triangle ADE is 27 cm².
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