Question

HELP!! MEDAL!!
An isosceles triangle has sides length 5,5,6. Find the measure, to the nearest degree, of each angle of the triangle.

Answer 1

IM HERE ONE MOMENT AS I RESEARCH

Answer 2

Sorry i didnt mean that to be in all caps

Answer 3

i didnt mind :)

Answer 4

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length b and the remaining side has length a. This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos (leg).

A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal. Another special case of an isosceles triangle is the isosceles right triangle.

The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as


h=sqrt(b^2-1/4a^2).
(1)


The area is therefore given by

A = 1/2ah
(2)

= 1/2asqrt(b^2-1/4a^2)
(3)

= 1/2a^2sqrt((b^2)/(a^2)-1/4).
(4)


The inradius of an isosceles triangle is given by


r=(a(sqrt(a^2+4h^2)-a))/(4h).
(5)


The mean of y is given by

<y> = int_(-a/2)^(a/2)int_0^([1-|x|/(a/2)]h)ydydx
(6)

= 1/6ah^2,
(7)


so the geometric centroid is

y^_ = (<y>)/A
(8)

= 1/3h,
(9)


or 2/3 the way from its vertex (Gearhart and Schulz 1990).
IsoscelesVertex
Considering the angle at the apex of the triangle and writing R instead of b, there is a surprisingly simple relationship between the area and vertex angle theta. As shown in the above diagram, simple trigonometry gives

h = Rcos(1/2theta)
(10)

x = Rsin(1/2theta),
(11)


so the area is

A = 1/2ah
(12)

= xh
(13)

= R^2cos(1/2theta)sin(1/2theta)
(14)

= 1/2R^2sintheta.

Answer 5

http://staff.tamhigh.org/wetzel/Ch8ansS14.pdf look at number 3

Answer 6

on the 3rd page

Answer 7

@isabellafiredust is right, although an easier way to do it is by doing inverse trigonometric ratios:
Start by finding the measure of a base angle of the triangle (we only need to find one because by definition, base angles of an isosceles triangle are congruent). You could use any inverse ratio since lengths of all sides are given, but I will use tangent:
the tangent ratio is the side opposite the base angle over the adjacent side, so in your case it’s 5/6.
Therefore, to find the measure of each of the base angles, we do:
measure of angle = \[\tan ^{-1}\times(6/10)\] = 30.9637565321, which is about 30.1, or 30. So, the measure of both base angles is 30 degrees.
By the Triangle Angle-Sum Theorem, we know that the sum of all three interior angles of a triangle is 180 degrees, so we could use this theorem to find the measure of the third angle (since two angle measures are now known):
30+30+x=180
60+x=180
x=180-60=120
The measure of the base angles of the isosceles triangle is 30 degrees, and the measure of the vertex angle is 120 degrees.
Happy studying! :)