Given that the measure of ∠x is 110°, and the measure of ∠y is 59°, find the measure of ∠z.

Mathematics

1 answer:

ch4aika [34]3 years ago

6 0

180-110=70
180-(70+59)=180-129=51
Z=51 im p sure

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An arch over a doorway is an arc that is 90 centimeters wide and 30 centimeters high. You are curious about the size of the enti

krek1111 [17]

Answer:

The length of the arc = $114.66\,cm$

And the length of the full circle: $306.3\, cm$

Step-by-step explanation:

To solve this problem we will be analyzing aright angle triangle that is formed when drawing the appropriate radius associated with the arc and the chord of a segment of circumference as depicted by the triangle shaded in yellow in the attached image.

Notice that the information given in this problem can be represented by:

1) the 90 cm width of the arc with the value of 2x (drawn in orange) in the drawing

2) the 30 cm of the arc's height with the value of y (drawn in green) in the drawing

In order to use the Pythagorean theorem in the little yellow triangle, notice that the hypotenuse is given by the radius R of the circle, and the two legs of the right angle triangle are given by "x" and "R-y"

So the Pythagoras theorem becomes:

$R^2=x^2+(R-y)^2\\R^2=x^2+R^2-2Ry+y^2\\0=x^2-2Ry+y^2\\2Ry=x^2+y^2\\R=\frac{x^2+y^2}{2y}$

Therefore, in our case, where x=45 cm, and y=30 cm, we have:

$R=\frac{x^2+y^2}{2y} \\R=\frac{45^2+30^2}{60}\,cm \\R=48.75\, cm$

Now, with the radius, we can calculate the length of the arc of circle if we know the subtended angle in radians:

length of arc = $R \,\theta$

The angle $\theta$ subtended by the arc can be obtained by doubling the angle with vertex at the circle's center in our yellow right angle triangle. This is done via the sine function: