In figure 4, AC is the diameter of a circle with centre O. Given that angle BAC = 37° and angle ACD = 46°…

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In figure 4, AC is the diameter of a circle with centre O. Given that angle BAC = 37° and angle ACD = 46°. Find the value of: a. angle x, b. angle y

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3. Figure 1 represents the height PR of a boy and his shadow RQ, 2.4m long on a flat horizontal ground.
This problem is incomplete. To find the distance PQ (the hypotenuse) and the angle of elevation of the sun from Q to P, we need either the height of the boy (PR) or the angle of elevation itself. Only the length of the shadow (RQ = 2.4 m) is provided.

4. In figure 4, AC is the diameter of a circle with centre O. Given that angle BAC = 37° and angle ACD = 46°. Find the value of:

a. angle x

Identify that AC is the diameter.
The angle subtended by a diameter at any point on the circumference is $90^\circ$ .
Therefore, $\angle ABC = 90^\circ$. $$ x = 90^\circ $$ The value of angle x is $\boxed{90^\circ}$.
- b. angle y
Identify that AC is the diameter.
The angle subtended by the diameter at point D on the circumference is $90^\circ$ .
Therefore, $\angle ADC = 90^\circ$.
Consider the sum of angles in triangle ADC.
$$ \angle DAC + \angle ACD + \angle ADC = 180^\circ $$
Substitute the known values.
$$ y + 46^\circ + 90^\circ = 180^\circ $$

Solve for y.

$$
y + 136^\circ = 180^\circ
$$
$$
y = 180^\circ - 136^\circ
$$
$$
y = 44^\circ
$$
The value of angle y is $\boxed{44^\circ}$.

c. angle DCB

Angle DCB is the sum of $\angle DCA$ and $\angle ACB$ .
We are given $\angle ACD = 46^\circ$ .
To find $\angle ACB$ , consider triangle ABC.
$$ \angle BAC + \angle ABC + \angle ACB = 180^\circ $$

Substitute the known values ( $\angle BAC = 37^\circ$ and $\angle ABC = 90^\circ$ ).

$$
37^\circ + 90^\circ + \angle ACB = 180^\circ
$$
$$
127^\circ + \angle ACB = 180^\circ
$$
$$
\angle ACB = 180^\circ - 127^\circ
$$
$$
\angle ACB = 53^\circ
$$

Calculate $\angle DCB$ .

$$
\angle DCB = \angle ACD + \angle ACB
$$
$$
\angle DCB = 46^\circ + 53^\circ
$$
$$
\angle DCB = 99^\circ
$$
The value of angle DCB is $\boxed{99^\circ}$.

In figure 4, AC is the diameter of a circle with centre O. Given that angle BAC = 37° and angle ACD = 46°. Find the value of: a. angle x, b. angle y

Step-by-step solution

Identify that AC is the diameter.

The angle subtended by a diameter at any point on the circumference is $90^\circ$ .

Identify that AC is the diameter.

The angle subtended by the diameter at point D on the circumference is $90^\circ$ .

Consider the sum of angles in triangle ADC.

Substitute the known values.

Solve for y.

Angle DCB is the sum of $\angle DCA$ and $\angle ACB$ .

To find $\angle ACB$ , consider triangle ABC.

Substitute the known values ( $\angle BAC = 37^\circ$ and $\angle ABC = 90^\circ$ ).

Calculate $\angle DCB$ .

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More Mathematics Questions

In figure 4, AC is the diameter of a circle with centre O. Given that angle BAC = 37° and angle ACD = 46°. Find the value of: a. angle x, b. angle y

Step-by-step solution

Identify that AC is the diameter.

The angle subtended by a diameter at any point on the circumference is 90∘90^\circ90∘.

Identify that AC is the diameter.

The angle subtended by the diameter at point D on the circumference is 90∘90^\circ90∘.

Consider the sum of angles in triangle ADC.

Substitute the known values.

Solve for y.

Angle DCB is the sum of ∠DCA\angle DCA∠DCA and ∠ACB\angle ACB∠ACB.

To find ∠ACB\angle ACB∠ACB, consider triangle ABC.

Substitute the known values (∠BAC=37∘\angle BAC = 37^\circ∠BAC=37∘ and ∠ABC=90∘\angle ABC = 90^\circ∠ABC=90∘).

Calculate ∠DCB\angle DCB∠DCB.

Still stuck on this one?

More Mathematics Questions

The angle subtended by a diameter at any point on the circumference is $90^\circ$ .

The angle subtended by the diameter at point D on the circumference is $90^\circ$ .

Angle DCB is the sum of $\angle DCA$ and $\angle ACB$ .

To find $\angle ACB$ , consider triangle ABC.

Substitute the known values ( $\angle BAC = 37^\circ$ and $\angle ABC = 90^\circ$ ).

Calculate $\angle DCB$ .