Question

The longest side of an isosceles obtuse triangle measures 20 centimeters. The other two side lengths are congruent but unknown.

What is the greatest possible whole-number value of the congruent side lengths?

A.9 cm
B.10 cm
C.14 cm
D.15 cm

Answer 1

c

Explanation:

Answer 2

Since 9+9=19<20 and 10+10=20=20, the triangles with sides 9, 9, 20 and 10, 10, 20 cannot exist.

Use the cosine theorem to check which option C or D is true:

1.

`20^2=14^2+14^2-2\cdot 14\cdot 14\cos \theta,\\ 400=196+196-392\cos \theta,\\ 8=-392\cos \theta,\\ \\ \cos \theta=-(8)/(392) =-(1)/(49)`.

You have that cosine of the greatest angle (opposite to the greatest side) is negative, this means that angle is obtuse.

2.

`20^2=15^2+15^2-2\cdot 15\cdot 15\cos \theta,\\ 400=225+225-450\cos \theta,\\ -50=-450\cos \theta,\\ \\ \cos \theta=(50)/(450) =(1)/(9)`.

Here you have that cosine of the greatest angle (opposite to the greatest side) is posiative, this means that angle is acute.

Answer: the greatest possible whole-number value of the congruent side lengths is 14 (choice C)