1) Find Angle C in the Diagram

[asy]

pair A,B,C;

B = (0,0);

A =(1,0);

C=(0.5,0.5*Tan(47));

draw(A--B--C--A);

label("$A$",A,SE);

label("$B$",B,SW);

label("$C$",C,N);

add(pathticks(A--C, 1, .5, 6, 2));

add(pathticks(C--B, 1, .5, 6, 2));

label("$47^{\circ}$",(0.16,0),N);

[/asy]

2) In the figure shown, ABCD is a square and Triangle CDE is equilateral. What is the degree measure of Angle CBE?

[asy]

import olympiad;

size(150);

pair D = (0, 10), C = (10, 10), E = (5, 18.66), B = (10, 0), A = (0, 0);

draw(A--B); draw(B--C); draw(C--D); draw(D--A); draw(D--E); draw(C--E); draw(B--E);

label("$A$", A, SW);

label("$B$", B, SE);

label("$E$", E, N);

label("$C$", C, NE);

label("$D$", D, NW);

[/asy]

3) In the diagram below, AM=BM=CM and Angle BMC + Angle A= 201 degrees. Find Angle B in degrees.

[asy]

size(5cm);

pair C = (0,0);

pair A = (25,0);

pair B = (20, 10);

pair M = (A+C)/2;

draw(C--A--B--cycle);

draw(B--M);

label("$C$", B, N);

label("$B$", C, SW);

label("$A$", A, SE);

label("$M$", M, S);

add(pathticks(A--M, 1, .5, 6, 30));

add(pathticks(B--M, 1, .5, 6, 30));

add(pathticks(C--M, 1, .5, 6, 30));

[/asy]

4)

A triangle whose side lengths are whole numbers has one side which measures 25 inches and a perimeter of 80 inches. What is the fewest number of inches that can be the length of one of the remaining sides?

5) In square units, what is the area of the gray triangle?

[asy]

size(150);

int gridsize = 7;

for (int i=-1; i<=gridsize; ++i) {

draw((i,-1)--(i,gridsize), mediumgray);

draw((-1,i)--(gridsize,i), mediumgray); }

draw((-1,0)--(gridsize,0), black+1bp, Arrows(6));

draw((0,-1)--(0, gridsize), black+1bp, Arrows(6));

label("$x$", (gridsize, 0), E);

label("$y$", (0, gridsize), N);

label("$0$", (0,0),SE, p=fontsize(8pt));

for (int i=1; i<=gridsize-1; ++i){

label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));

label("$"+string(i)+"$",(0,i),W, p=fontsize(8pt));}

defaultpen(fontsize(10pt));

filldraw((5,6)--(3,0)--(1,3)--cycle,gray,black);

[/asy]

6) In the diagram, Triangle BDF and Triangle ECF have the same area. If DB=2, BA=3 and AE=4 find the length of Line Segment EC.

[asy]

size(5cm);

pair A,B,C,D,EE,F;

A=(0,0);

B=(0,3);

C=(20/3,0);

D=(0,5);

EE=(4,0);

F=extension(B,C,D,EE);

draw(B--A--C--B--D--EE--A);

draw(rightanglemark(C,A,B,10));

label("$B$",B,W);

label("$A$",A,SW);

label("$D$",D,N);

label("$E$",EE,S);

label("$C$",C,S);

label("$F$",F,NE);

[/asy]

7) Shaina has one stick of length *a* cm and another of length *b* cm, where *a *isnt equal to *b.* She needs a third stick with length strictly between 8 cm and 26 cm to make the third side of a triangle. What is the product *ab*?

8)The points of this 3-by-3 grid are equally spaced horizontally and vertically. How many different sets of three points of this grid can be the three vertices of an isosceles triangle?

[asy]

size(50);

dot((0,0));

dot((10,0));

dot((20,0));

dot((0,10));

dot((10,10));

dot((20,10));

dot((0,20));

dot((10,20));

dot((20,20));

[/asy]

9)Two triangles are considered the same if they have the same three side lengths. Find all possible values of *P* such that there is only 1 possible triangle with integer side lengths and perimeter *P.*

Enter your answer as a comma-separated list.

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Thanks!

The digrams are all in asyptode code

Rudram592 Feb 9, 2019