measures of line with tangents?

[isu23pink]

i really need help with these four questions.
i have no idea what i did wrong:

for #6: i did
m Xdegreese = 1/2 (x+y) = 100 [WRONG]

for #7: i did
360- 250
1/2 110
x=55 [WRONG]

for #14 i did:
90= 1/2 (100 + x)
= (50+1/2x)
x=20 [WRONG]

 

[galactus]

#14. \(\displaystyle 90=\frac{1}{2}(100+x), \;\ x=80\)

 

[Mrspi]

i really need help with these four questions.
i have no idea what i did wrong:

for #6: i did
m Xdegreese = 1/2 (x+y) = 100 [WRONG]

for #7: i did
360- 250
1/2 110
x=55 [WRONG]

for #14 i did:
90= 1/2 (100 + x)
= (50+1/2x)
x=20 [WRONG]

For #6, if two secants, two tangents, or a secant and a tangent intersect OUTSIDE a circle, then the measure of the angle formed is (1/2) * (difference of the two arcs).

So, x = (1/2)*(140 - 60)

See if that works better for you.

For #7, ONE of the arcs intercepted by the angle is 250 degrees. The OTHER arc is the remainder of the circle, or 360 - 250, or 110 degrees. The measure of the angle is half the difference of the intercepted arcs, or (1/2)*(250 - 110)

 

[Mrspi]

i really need help with these four questions.
i have no idea what i did wrong:

for #6: i did
m Xdegreese = 1/2 (x+y) = 100 [WRONG]

for #7: i did
360- 250
1/2 110
x=55 [WRONG]

for #14 i did:
90= 1/2 (100 + x)
= (50+1/2x)
x=20 [WRONG]

For #8, the angle between a tangent and a radius drawn to the point of tangency is a right angle, with a measure of 90. That should enable you to find y easily.

Then, you know that one angle of the quadrilateral is 55 degrees, and two of the angles (where a radius meets a tangent at a point of tangency) are right angles. Should be easy to find x.

 

[isu23pink]

thank you guys for your help. but can you explain #8 to be better please?

 

[Mrspi]

thank you guys for your help. but can you explain #8 to be better please?

Look at your diagram for #8....you have a "diamond-shaped" quadrilateral. One of the angles is GIVEN to be 55 degrees. TWO of the angles are formed by radii of the circle which are drawn to the two points of tangency. Each of those angles is a right angle, and each of those has a measure of 90 degrees. The fourth angle is the one whose measure x you are to find.

What do you know about the sum of the angles of any quadrilateral? If you know the measures of three of the angles of a quadrilateral, it should be a matter of some simple arithmetic to find the measure of the fourth angle.

 

[fasteddie65]

Another way to see #8 is to realize that the 55° angle is formed by two tangents intersecting outside the circle. The measure of such an angle is 1/2 the difference of the intercepted arcs. One arc, intercepted by the central angle of x°, is also x° in measure, so the major arc with the same endpoints must be (360 - x)°. The angle of 55°, then, is (1/2)[(360 - x)° - x°] = (1/2)(360 - 2x)° = (180 - x)°. If 180 - x = 55, then x = 180 - 55 = 125.
The y° angle must be a right angle, because a radius drawn to a point of tangency is perpendicular at that point.