# Digital SAT Math Practice Questions -Medium : Circles

## SAT MAth Practice questions – all topics

• Geometry and Trigonometry Weightage: 15% Questions: 5-7
• Area and volume
• Lines, angles, and triangles
• Right triangles and trigonometry
• Circles

## SAT MAth and English  – full syllabus practice tests

[calc]  Question  Medium

A circle in the $$x y$$-plane has its center at $$(3,5)$$ and has a radius of 6 . What is an equation of the circle?
A) $$(x-3)^2+(y-5)^2=6$$
B) $$(x+3)^2+(y+5)^2=6$$
C) $$(x-3)^2+(y-5)^2=36$$
D) $$(x+3)^2+(y+5)^2=36$$

Ans: C

The standard form for the equation of a circle with center $$(h, k)$$ and radius $$r$$ is:

$(x – h)^2 + (y – k)^2 = r^2$

Given the center $$(3, 5)$$ and radius $$6$$, we substitute $$h = 3$$, $$k = 5$$, and $$r = 6$$:

$(x – 3)^2 + (y – 5)^2 = 6^2$

Simplify $$6^2$$:

$(x – 3)^2 + (y – 5)^2 = 36$

$\boxed{C}$

[calc]  Question  medium

Line $$k$$ is tangent to the circle with center $$C$$ at point $$A$$, as shown. What is the slope of line $$k$$ ?

A. -2
B. $$-\frac{1}{2}$$
C. $$\frac{1}{2}$$
D. 2

Ans:A

To find the slope of line $$k$$, which is tangent to the circle at point $$A$$, we need to find the slope of the line passing through points $$C$$ and $$A$$. The slope of a line passing through two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by:

$m = \frac{{y_2 – y_1}}{{x_2 – x_1}}$

Given points $$C(-5, -4)$$ and $$A(-7, -5)$$:

$m = \frac{{-5 – (-4)}}{{-7 – (-5)}}$
$m = \frac{{-1}}{{-7 + 5}}$
$m = \frac{{-1}}{{-2}}$
$m = \frac{1}{2}$

So, the slope of line $$k$$ is $$\frac{1}{2}$$, which corresponds to option C.

[Calc]  Question  medium

A circle has been divided into three nonoverlapping regions: I, II, and III. The area of region I is $$4\pi$$ square centimeters ($$cm^{2}$$), the area of region II is $$12\pi$$ $$cm^{2}$$, and the area of region III is $$16\pi$$ $$cm^{2}$$. If a point in the circle is selected at random, what is the probability of selecting a point that does not lie in region II? (Express your answer as a decimal or fraction, not as a percent.)

Ans: 5/8, .625

To find the probability of selecting a point that does not lie in region II, we need to find the total area of the circle and subtract the area of region II, then divide by the total area of the circle.

Given:
Area of region I = $$4\pi \text{ cm}^2$$
Area of region II = $$12\pi \text{ cm}^2$$
Area of region III = $$16\pi \text{ cm}^2$$

The total area of the circle is the sum of the areas of all three regions:
$4\pi + 12\pi + 16\pi = 32\pi \text{ cm}^2$

So, the probability of selecting a point that does not lie in region II is:

$\frac{4\pi + 16\pi}{32\pi} = \frac{20\pi}{32\pi} = \frac{5}{8}$

Therefore, the probability is $$\frac{5}{8}$$.

[No- Calc]  Question  Medium

$$x^{2}-10x+y^{2}+6y=2$$

The graph in the xy-plane of the equation above is a circle. What are the coordinates of the center of the circle?

A) (-5,-3)
B) (-5,3)
C) (5,-3)
D) (5,3)

Ans: C

$$x^2-10x+y^2+6y=2$$ represents a circle equation in the form $$(x – h)^2 + (y – k)^2 = r^2$$, where $$(h, k)$$ is the center of the circle.

For $$x$$, we complete the square by adding $$(10/2)^2 = 25$$ inside the parenthesis:

$x^2 – 10x + 25 + y^2 + 6y = 2 + 25$

$x^2 – 10x + 25 + y^2 + 6y + 9 = 27$

$(x – 5)^2 + (y + 3)^2 = 27$

Comparing this to the standard form $$(x – h)^2 + (y – k)^2 = r^2$$, we see that the center of the circle is $$(h, k) = (5, -3)$$.

So, the coordinates of the center of the circle are C) $$(5, -3)$$.

[Calc]  Question Medium

$x^2+y^2-16 x-4 y+32=0$

In the $$x y$$-plane, the graph of the given equation is a circle. What is the length of the radius of this circle?
A) 2
B) 6
C) 8
D) 36

B

The given equation is:
$x^2 + y^2 – 16x – 4y + 32 = 0$

Rewrite the equation in standard form of a circle $$(x-h)^2 + (y-k)^2 = r^2$$:

First, complete the square for $$x$$ and $$y$$.

For $$x$$:
$x^2 – 16x$
Complete the square:
$x^2 – 16x + 64 – 64$
$(x-8)^2 – 64$

For $$y$$:
$y^2 – 4y$
Complete the square:
$y^2 – 4y + 4 – 4$
$(y-2)^2 – 4$

Rewrite the equation with the completed squares:
$(x-8)^2 – 64 + (y-2)^2 – 4 + 32 = 0$
$(x-8)^2 + (y-2)^2 – 36 = 0$
$(x-8)^2 + (y-2)^2 = 36$

The equation $$(x-8)^2 + (y-2)^2 = 36$$ represents a circle with radius $$\sqrt{36} = 6$$.

$\boxed{B}$

[Calc]  Question   Medium

What is the radius of the circle in the $x y$-plane with equation $(x-9)^2+(y-3)=64$ ?
A) 64
B) 9
C) 8
D) 3

C

[Calc]  Question  Medium

In the given figure, $\theta$ is an angle. If $\sin \theta=\frac{\sqrt{3}}{2}$, what is $\cos \theta$ ?
A) $\frac{\sqrt{3}}{2}$
B) $\frac{1}{2}$
C) $-\frac{1}{2}$
D) $-\frac{\sqrt{3}}{2}$

C

Question

In the $$xy$$-plane, the points (-3, 10) and (3, 10) are endpoints of the diameter of a circle. Which equation represents this circle?

1. $$x$$2+($$y$$-10)2=9
2. ($$x$$+3)2+$$y$$2=36
3. ($$x$$+3)2+($$y$$-10)2=9
4. ($$x$$+3)2+($$y$$-10)2=36

A

Question

In the figure shown, C is the center of the circle and $\overline{AB}$ is tangent to the circle at A. Which of the following is true about the measure of angle BAC? 4.5

1. The measure is less than $90\degree$
2. The measure is greater than $90\degree$
3. The measure is equal to $90\degree$
4. It cannot be determined whether the measure is less than, greater than, or equal to $90\degree$

C

Question

A circle has center C at (1,1) and radius 2. Which of the following is an equation of this circle?

1. ($$x$$+1)2+($$y$$+1)2=2
2. ($$x$$+1)2+($$y$$+1)2=4
3. ($$x$$-1)2+($$y$$-1)2=2
4. ($$x$$-1)2+($$y$$-1)2=4

D

Question

In the $x y$-plane, a circle with radius 5 has center $(-8,6)$. Which of the following is an equation of the circle?
A. $(x-8)^2+(y+6)^2=25$
B. $(x+8)^2+(y-6)^2=25$
C. $(x-8)^2+(y+6)^2=5$
D. $(x+8)^2+(y-6)^2=5$

Ans: B

Questions

Triangle $F G H$ is inscribed in the circle above. If arc $\widehat{F G}$ is congruent to arc $\widehat{G H}$, and the measure of $\angle \mathrm{G}$ is $30^{\circ}$, what is the measure of $\angle \mathrm{H}$ ?
A. $30^{\circ}$
B. $60^{\circ}$
C. $75^{\circ}$
D. $120^{\circ}$

Ans: C

Question

In the $x y$-plane, the points $(2,4)$ and $(-2,-4)$ are the endpoints of a diameter of a circle, Which of the following is an equation of the circle?
A. $(x-2)^2+(y+4)^2=80$
B. $(x-2)^2+(y+4)^2=20$
C. $x^2+y^2=80$
D. $x^2+y^2=20$