Theorems: Circumscribed and Inscribed Circles

Theorems: Circumscribed and Inscribed Circles. Practice questions to deepen understanding of theorems on circumscribed and inscribed circles. Online math practice with full solutions and step-by-step explanations.

Circumscribed and inscribed circles practice — circumscribed and inscribed circles in triangles and quadrilaterals, conditions for inscription, regular polygons. Detailed explanations.

25 questions

Question 1
4.00 pts

📐 inscribed circle:
ABC given: AB = 13 cm, BC = 14 cm, AC = 15 cm
radius ( two ).

ABC131514r

Explanation:

💡 explanation :

1: calculation half perimeter 📏

s = (a + b + c) / 2

s = (13 + 14 + 15) / 2

s = 21 cm

2: calculation area ( ) 🔢

Question 2
4.00 pts

🔺 Right triangle:
In right triangle ABC, with the right angle at C, AB = 10 cm, AC = 6 cm, and BC = 8 cm.
What is the radius of the incircle?

Explanation:

💡 Detailed explanation:

For a right triangle, the inradius is:

r = (a + b − c) / 2, where c is the hypotenuse.

Here: r = (6 + 8 − 10) / 2 = 4 / 2 = 2 cm.

Question 3
4.00 pts

⭐ equal side:
equal side length side 12 cm
radius ( one).

ABC121212r

Explanation:

💡 explanation :

1: equal side 🔺

equal side:

r = a√3 / 6

2: 🔢

Question 4
4.00 pts

📊 :
given ABC area 60 cm² 40 cm
radius ?

ABCS = 60 cm²perimeter = 40 cmr = ?

Explanation:

💡 explanation :

1: given? 📝

✓ area of the triangle: S = 60 cm²

✓ perimeter : P = 40 cm

2: calculation half perimeter 📏

Question 5
4.00 pts

🔻 isosceles triangle:
ABC isosceles: AB = AC = 10 cm, BC = 12 cm
radius ( one).

ABC101012r

Explanation:

💡 explanation :

1: calculation altitude 📐

Question 6
4.00 pts

⭕ inscribed in a circle:
ABC inscribed in a circle radius 5 cm
angle A = 30°
length side BC.

30°ABCR = 5BC = ?

Explanation:

💡 explanation :

1: theorem 📐

altitude -A base BC

bisects BC two : 6 cm one

by/according to Pythagorean theorem:
h = √(10² - 6²)
h = √(100 - 36)
h = √64
h = 8 cm
theorem :

a = 2R · sin(A)

:
a = side angle A
R = radius

2: 🔢

Question 7
4.00 pts

📐 inscribed in a circle:
ABC (angle -C)
given: AB = 10 cm
radius ?

ACB10 cmR = ?angle diameter

Explanation:

💡 explanation :

1: theorem ! 🎯

angle diameter
angle (90°)

2: 💭

Question 8
4.00 pts

⭐ equal side inscribed in a circle:
equal side ABC inscribed in a circle radius 6 cm
length side ( one).

ABCR = 6a = ?

Explanation:

💡 explanation :

1: equal side 🔺

equal side inscribed in a circle:

a = R√3

2: 🔢

Question 9
4.00 pts

🔷 - angle 45°:
ABC inscribed in a circle
given: BC = 8 cm, angle A = 45°
radius ( one).

45°ABCBC = 8R = ?

Explanation:

💡 explanation :

1: theorem 📐

theorem :

BC / sin(A) = 2R

2: 🔢

Question 10
4.00 pts

⭕ angle in semicircle:
ABC inscribed in a circle radius R = 7 cm
angle A = 90° (angle )
length side BC?

BAC90°R = 7BC = ?

Explanation:

💡 explanation :

1: theorem angle 📐

angle diameter
equal -90°

⬇️

angle A = 90°
BC = diameter

2: between radius diameter 🔢

Question 11
4.00 pts

🔷 inscribed in a circle:
ABCD inscribed in a circle
given: ∠A = 75°, ∠B = 110°, ∠C = 105°
angle ∠D.

ABCD75°110°105°?

Explanation:

💡 explanation :

1: theorem opposite angles 📐

inscribed in a circle:

opposite angles = 180°

∠A + ∠C = 180°
∠B + ∠D = 180°

2: angles A -C ✓

Question 12
4.00 pts

❓ :
?
ABCD angles:
∠A = 80°, ∠B = 95°, ∠C = 100°, ∠D = 85°

?ABCD80°95°100°85°

Explanation:

💡 explanation :

1: 📋

:

inscribed in a circle
⬇️
opposite angles = 180°

2: (A -C) 🔍

Question 13
4.00 pts

🔶 inscribed in a circle:
ABCD inscribed in a circle
(AB parallel -CD)
theorem Correct?

ABCDAB ∥ CD

Explanation:

💡 explanation :

1: cyclic quadrilateral 📐

inscribed in a circle:

∠A + ∠C = 180°
∠B + ∠D = 180°

2: 📏

Question 14
4.00 pts

🔢 finding :
ABCD inscribed in a circle
given: ∠A = (2x + 10)°, ∠C = (3x - 20)°
x.

ABCD(2x+10)°(3x-20)°180°

Explanation:

💡 explanation :

1: 📝

angles A -C

⬇️

∠A + ∠C = 180°

(2x + 10) + (3x - 20) = 180

2: 🔢

Question 15
4.00 pts

📏 circumscribes a circle:
ABCD circumscribes a circle
given: AB = 5 ", BC = 7 ", CD = 6 "
length side DA.

ABCD576? side

Explanation:

💡 explanation :

1: theorem side 📐

circumscribes a circle:

side equal

AB + CD = BC + DA

2: identification side 🔍

Question 16
4.00 pts

❓ :
circumscribes a circle?
ABCD side:
AB = 8 cm, BC = 6 cm, CD = 10 cm, DA = 4 cm

?ABCD86104AB + CD = ? | BC + DA = ?

Explanation:

💡 explanation :

1: 📋

:

circumscribes a circle
⬇️
side equal

2: calculation side 🔢

Question 17
4.00 pts

🔢 finding - :
ABCD circumscribes a circle
given: AB = 2x, BC = x + 3, CD = 2x - 2, DA = x + 1
x.

ABCD2xx+32x-2x+1AB + CD = BC + DA

Explanation:

💡 explanation :

1: 📝

circumscribes a circle:

AB + CD = BC + DA

2x + (2x - 2) = (x + 3) + (x + 1)

2: two 🔢

Question 18
4.00 pts

⬡ regular hexagon inscribed in a circle:
regular hexagon length side 6 cm inscribed in a circle
radius ?

AB6 cmR = ?

Explanation:

💡 explanation :

1: special regular hexagon ⬡

regular hexagon:

radius
=
length side

R = a

2: explanation 💭

Question 19
4.00 pts

◼️ square inscribed in a circle:
square ABCD inscribed in a circle radius 5 cm
length side square ( one).

ABR = 5a = ?

Explanation:

💡 explanation :

1: between radius No 📐

square inscribed in a circle:

= diameter

d = 2R

2: calculation diameter 🔢

Question 20
4.00 pts

⭐ ratio between circles:
equal side length side 12 cm
between radius
radius ?

RrR : r = ?

Explanation:

💡 explanation :

1: equal side 📐

radius :
R = a√3 / 3 = a / √3

radius :
r = a√3 / 6 = a / (2√3)

2: calculation R / r 🔢

Question 21
4.00 pts

❓ one:
one?

Pinfinitely many one
Explanation:

💡 explanation :

1: ⭕

=

metres
(r)
given (O)

2: one 🔍

🔹
Question 22
4.00 pts

🔵 two :
two A -B?

AB midpoint midpoint
Explanation:

💡 explanation :

1: 📐

A -B

⬇️

must equal-
-A -B

2: metres 🔍

🔹 - -A -B
<
Question 23
4.00 pts

❌ three :
three one
three ?

ABC 3
Explanation:

💡 explanation :

1: No possible? 🤔

=


(No because )

2: 🔍

Question 24
4.00 pts

📍 finding :
three A, B, C one
Yes?

Obecause midpoint O

Explanation:

💡 explanation :

1: 📐

(O)

⬇️

must equal-
-A, B, -C

OA = OB = OC = R

2: metres 🔍

Question 25
4.00 pts

✨ theorem :
one
one ?

ABC one ! 3
Explanation:

💡 explanation :

1: theorem 📚

three
one
one

2: 3? 🤔

← Back to all exams