Find the truth - Geometry Edition

Geometry Level 5

(I) a, b and c are sides of a triangle, then a+b>c.

(2) If ABC is a right triangle with integer sides, then its inradius is also an integer.

(3) l, m and n are 3 lines such that no 2 are parallel to each other, then there are 2 different points on n which are equedistant to l and m.

(4)If in a triangle ABC, ${ AB }^{ 2 }+{ CB }^{ 2 }={ AC }^{ 2 }$ , then ABC is right angled.

(5) Area of triangle ABC is greater than that of PQR, then Perimeter of ABC is greater than PQR.

(6) If ABC and PQR have equal area, AC=PR and AB=PQ, then they both are congruent.

Add the serial numbers of all the true statements from these.

For e.g. If (3), (4) and (6) are true, the answer is 3+4+6=13, if none are true answer is 0.

12 0 7 13 6 8 15 21

1 solution

Archit Boobna
Mar 29, 2015

(I) TRUE - Triangle inequality law

(2) TRUE - $Inradius\quad in\quad a\quad a-b-c\quad right\quad triangle\quad is\quad given\quad by\\ \\ r=\frac { a+b-c }{ 2 } \\ If\quad both\quad a\quad and\quad b\quad are\quad odd,\quad then\quad c\quad is\quad even.\\ So\quad a+b-c\quad is\quad even,\quad so\quad r\quad is\quad an\quad integer.\\ \\ If\quad both\quad a\quad and\quad b\quad are\quad even,\quad then\quad c\quad is\quad even.\\ So\quad a+b-c\quad is\quad even,\quad so\quad r\quad is\quad an\quad integer.\\ \\ If\quad one\quad of\quad a\quad and\quad b\quad is\quad odd,\quad then\quad c\quad is\quad odd.\\ So\quad a+b-c\quad is\quad even,\quad so\quad r\quad is\quad an\quad integer.\quad$

(3) FALSE - This is false when l, m and n are concurrent and n is not the angle bisector of l and m.

(4) TRUE - Can be directly proved by cosine rule.

(5) FALSE - Let ABC be 1, 1, 1 PQR be 2, 2, 3.99

(6) FALSE - Let ABC be 5,5,6 PQR be 5,5,8

Find the truth - Geometry Edition

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