MATH SOLVE

4 months ago

Q:
# 48 POINTS HELP ME WITH ONE GEOMETRY QUESTION I'M FAILING!!!!!!!!HI, PLEASE HELP A BROTHA OUT I HAVE A 36 AT THE MOMENT AND I AM VERY CLOSE TO GRADUATING. HOWEVER, I'M ONLY FAILING GEOMETRY. GRADUATION IS IN 48 DAYS HELP MEEXPLAIN IT IF YOU COULD, BUT IF YOU CAN'T THAT'S ALRIGHT

Accepted Solution

A:

Looks like the line through [tex]E[/tex] and [tex]D[/tex] is tangent to the circle. It also looks like the [tex]70^\circ[/tex] is supposed to represent the measure of the arc [tex]CD[/tex].

So we know the measure of the central angle subtended by the arc is also [tex]70^\circ[/tex]. What this means is, if you were to draw a line from [tex]D[/tex] to the circle's center (call it [tex]O[/tex]), and another from the center to [tex]C[/tex], these two lines would form an angle of [tex]70^\circ[/tex] in the "direction" of the minor arc [tex]CD[/tex].

Now [tex]COD[/tex] forms an isosceles triangle. We know that because the line segments [tex]DO[/tex] and [tex]CO[/tex] both have lengths equal to the circle's radius. Because [tex]\Delta COD[/tex] is isosceles, the angles in the triangle that don't coincide with the central angle we already know about must have the same measures. We call them both [tex]x^\circ[/tex].

The interior angles of a triangle always add up to [tex]180^\circ[/tex], so we can easily find [tex]x[/tex]:

[tex]180=x+x+70\implies x=55[/tex]

Finally, the line segments [tex]DO[/tex] and [tex]DE[/tex] are perpendicular; this is true because [tex]DE[/tex] is tangent to the circle. This means the angles [tex]x[/tex] and the one indicated by "?" are complementary and add to [tex]90^\circ[/tex].

So, [tex]x+?=90\implies ?=90-55=35[/tex].

So we know the measure of the central angle subtended by the arc is also [tex]70^\circ[/tex]. What this means is, if you were to draw a line from [tex]D[/tex] to the circle's center (call it [tex]O[/tex]), and another from the center to [tex]C[/tex], these two lines would form an angle of [tex]70^\circ[/tex] in the "direction" of the minor arc [tex]CD[/tex].

Now [tex]COD[/tex] forms an isosceles triangle. We know that because the line segments [tex]DO[/tex] and [tex]CO[/tex] both have lengths equal to the circle's radius. Because [tex]\Delta COD[/tex] is isosceles, the angles in the triangle that don't coincide with the central angle we already know about must have the same measures. We call them both [tex]x^\circ[/tex].

The interior angles of a triangle always add up to [tex]180^\circ[/tex], so we can easily find [tex]x[/tex]:

[tex]180=x+x+70\implies x=55[/tex]

Finally, the line segments [tex]DO[/tex] and [tex]DE[/tex] are perpendicular; this is true because [tex]DE[/tex] is tangent to the circle. This means the angles [tex]x[/tex] and the one indicated by "?" are complementary and add to [tex]90^\circ[/tex].

So, [tex]x+?=90\implies ?=90-55=35[/tex].