Triangles questions set 2

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1)Find the number of isosceles obtuse angled triangle such that their perimeter is 2020


2)ABCD is a quadrilateral with AD k BC. If the angle bi-
sector of ∠DAB intersects CD at E, and BE bisects ∠ABC, prove that
AB = AD + BC.(CHINA)

3)Given that Right ∆ABC has a perimeter of 30 cm and an area of 30 cm2. Find the lengths of its three sides. 
Ans
8,15,17


4)Points M and N are the midpoints of sides PA and PB of ∆PAB. As P moves along a line that is parallel to side AB , how many of the four quantities listed below change?

(a) the length of the segment MN

(b) the perimeter of ∆PAB

(c) the area of ∆PAB

(d) the area of trapezium ABNM

AMC 10 2010

Ans B


5)∆ABC is an isosceles triangle with AB = AC = 2.There are 20 points P₁, P₂, . . . , P₂₀on the side BC. Write mi = APᵢ² +BPᵢ· PᵢC (i = 1, 2, . . . , 20), find the value of m₁ + m₂ + · · · + m₂₀

6)

[asy] unitsize(27); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E,F,X,Y,Z; A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); X=intersectionpoint(A--D,C--F); Y=intersectionpoint(B--E,A--D); Z=intersectionpoint(B--E,C--F); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,S); label("$E$",E,NE); label("$F$",F,NW); label("$N_1$",X,NE); label("$N_2$",Y,WNW); label("$N_3$",Z,S); [/asy]

In the figure, CD, AE and BF are one-third of their respective sides. It follows that              

            AN₂:N₂N₁:N₁D=3:3:1 

and similarly for lines BE and CF. Then the area of trianglei N₁,N₂,N₃

 is x times of ∆ABC

(ASHME)


                       Level-2

1)Let ABC be an acute-angled, not equilateral triangle, where vertex A lies on the perpendicular bisector of the segment HO, joining the orthocentre H to the circumcentre O. Determine all possible values for the measure of angle A.

(U.S.A. - 1989 IMO Shortlist)

2)Let ABC be an acute-angled, not equilateral triangle, where vertex A lies on the perpendicular bisector of the segment HO, joining the orthocentre H to the circumcentre O. Determine all possible values for the measure of angle A.
(U.S.A. - 1989 IMO Shortlist)


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