Answer: ABCD is a rhombus having AC and BD its diagonals. The diagonals of a rhombus perpendicular bisect each other. AOC is a right-angled triangle. In right triangle AOC, By using Pythagoras...
Answers: (i) True Given, ∆ABC ~ ∆DEF ∠???????????? = ∠???????????? ∠???? = ∠???? (∠???????????? ~ ∆????????????) By AA criterion, ∆ABP and ∆DEQ $\frac{A B}{D E}=\frac{A P}{D Q}$...
and AC = 8 cm and in a DEF , DF = 9 cm D = and DE= 12 cm, then ABC ~ DEF. (ii) the polygon formed by joining the midpoints of the sides of a quadrilateral is a rhombus.Answers: (i) False In ∆ABC, AB = 6 cm ∠???? = 450 ???????? = 8 ???????? I???? ∆????????????, ???????? = 9 ???????? ∠???? = 450 ???????? = 12 ???????? ∆???????????? ~ ∆???????????? (ii) False...
Answers: (i) False If two triangles are similar, their corresponding angles are equal and their corresponding sides are proportional. (ii) True ABC is a triangle with M, N DE is...
Answers: (i) False Two rectangles are similar if their corresponding sides are proportional. (ii) True Two circles of any radii are similar to each other.
Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 20 ???????? ???????? = 21 ???????? In right...
. If p, q and r are the lengths of AM, MB and BC respectively then express the length of MN of terms of P, q and r.Answer: In ∆AMN and ∆ABC, ∠???????????? = ∠???????????? =$76^{\circ}$ ∠???? = ∠???? (????????????????????????) By AA similarity criterion, ∆AMN ~ ∆ABC If two triangles...
Answer: Let, DC = x BD = a - x Using angle bisector there in ∆ ABC, $\frac{A B}{A C}=\frac{B D}{D C}$ $\frac{c}{b}=\frac{a-x}{x}$ $c x=a b-b x$ $x(b+c)=a b$ $x=\frac{a...
Answer: In right-angled triangle SOW, Using Pythagoras theorem, ????????2 = ????????2 + ????????2 => 352 + 122 => 1225 + 144 => 1369 ???????? = 37 ???? Hence,...
Answer: The altitude drawn from the vertex opposite to the non-equal side bisects the non-equal side. ABC is an isosceles triangle having equal sides AB and BC. The altitude drawn from the vertex...
Answer: When two triangles are similar, then the ratios of the lengths of their corresponding sides are proportional. ∆BMP ~ ∆CNR $\frac{B M}{C N}=\frac{B P}{C R}=\frac{M P}{N R}$ $\quad \frac{B...
Answer: Given, AM : MB = 1 : 2 $\frac{M B}{A M}=\frac{2}{1}$ Adding 1 to both sides, $\frac{M B}{A M}+1=\frac{2}{1}+1$ $\frac{M_{B}+A M}{A M}=\frac{2+1}{1}$ $\frac{A B}{A...
and 
. If ∆DEF ~ ∆GHK then find the measures of ∠F.Answer: If two triangles are similar then the corresponding angles of the two triangles are equal. ∆DEF ~ ∆GHK ∴ ∠???? = ∠???? = 570 In ∆ DEF Using the a????????????????...
Answer: ABCD is a rhombus. The diagonals of a rhombus perpendicularly bisect each other. ∠???????????? = 900 ???????? = 12 ???????? ???????? = 5 ???????? In right triangle AOB,...
Answer: We know that the altitude of an equilateral triangle bisects the side on which it stands and forms right angled triangles with the remaining sides. ABC is an...
Answer: If two triangles are similar, then the ratio of their areas is equal to the squares of their corresponding sides. $\frac{\text { area of } smaller triangle}{\text { area of } larger...
Answer: In ∆ AOB and COD ∠???????????? = ∠???????????? (???????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????) ∠???????????? = ∠????????????...
Answer: Given, ∆ ABC ~ ∆ DEF If two triangles are similar then the ratio of their areas is equal to the ratio of the squares of their corresponding sides. $\frac{\text { area }(\triangle A B...
Answer: The altitude of an equilateral triangle bisects the side on which it stands and forms right angled triangles with the remaining sides. ABC is an equilateral...
Answer: Let AB be A ladder and B is the window at 8 m above the ground C. In right triangle ABC By using Pythagoras theorem, ????????2 = ????????2 + ????????2 102 = 82 +...
Answer: In ∆ADE and ∆ABC, ∠???????????? = ∠???????????? (???????????????????????????????????????????????????? ???????????????????????? ???????? ???????? ∥ ????????)...
Answer: When two triangles are similar, then the ratios of the lengths of their corresponding sides are equal. ∆ABC ~ ∆DEF $\therefore \frac{A B}{D E}=\frac{B C}{E F}$ $\frac{A B}{2 A B}=\frac{6}{E...
, PR = 9cm ∠???? = and PQ = 4.5 cm. Show that ∆ ABC ~ ∆ PQR and state that similarity criterion.Answer: In ∆ABC and ∆PQR ∠???? = ∠???? = 700 $\frac{A B}{P Q}=\frac{A C}{P R}$ $\frac{3}{4.5}=\frac{6}{9}$ $\frac{1}{1.5}=\frac{1}{1.5}$ By SAS similarity criterion, ∆ABC ~...
Answer: Using midpoint theorem, The segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of...
Converse of Pythagoras theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
Pythagoras theorem: The square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, the hypotenuse is the longest side and it’s always opposite the right angle.
SAS-similarity criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar.
SSS-similarity criterion for similarity of triangles: If the corresponding sides of two triangles are proportional then their corresponding angles are equal, and hence the two triangles are...
AA-similarity criterion: If two angles are correspondingly equal to the two angles of another triangle, then the two triangles are similar.
AAA-similarity criterion: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and hence the triangles are similar.
Midpoint theorem: The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is equal to one half of the third side.
Thale’s theorem: If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.
Basic proportionality theorem: If a line is draw parallel to one side of a triangle intersect the other two sides, then it divides the other two sides in the same ratio.
Answer: The two triangles are similar if and only if The corresponding sides are in proportion. The corresponding angles are equal.