5) A (4y +14)^ 105^ C (7y+1)^ (7x+1)^ B An inscribed quadrilateral in a circle is shown, which statements are correct? 4))) A) x = 13^ B) y = 15^ C) \angle A = 74^ D) \angle B = 75^ E) \angle C = 108^
The Correct Answer and Explanation is:
To solve this, we analyze the properties of an inscribed quadrilateral in a circle.
Key Property:
In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the opposite angles are supplementary, meaning they add up to 180°.
From the image, we are given:
- ∠C = 105°
- ∠A = (4y + 14)°
- ∠B = (7x + 1)°
- ∠D = (7y + 1)°
Let’s now apply the rule about opposite angles.
Step 1: Use ∠A + ∠C = 180°
(4y+14)+105=180⇒4y+119=180⇒4y=61⇒y=15.25(4y + 14) + 105 = 180 \Rightarrow 4y + 119 = 180 \Rightarrow 4y = 61 \Rightarrow y = 15.25(4y+14)+105=180⇒4y+119=180⇒4y=61⇒y=15.25
This is not equal to 15, so option B is incorrect.
Step 2: Use ∠B + ∠D = 180°
(7x+1)+(7y+1)=180⇒7x+7y+2=180(7x + 1) + (7y + 1) = 180 \Rightarrow 7x + 7y + 2 = 180(7x+1)+(7y+1)=180⇒7x+7y+2=180
Now substitute y=15.25y = 15.25y=15.25:7x+7(15.25)+2=1807x+106.75+2=1807x=180−108.75=71.25x=10.187x + 7(15.25) + 2 = 180 \\ 7x + 106.75 + 2 = 180 \\ 7x = 180 – 108.75 = 71.25 \\ x = 10.187x+7(15.25)+2=1807x+106.75+2=1807x=180−108.75=71.25x=10.18
So x ≠ 13, meaning option A is incorrect.
Step 3: Calculate angles using values
- ∠A = 4y + 14 = 4(15.25) + 14 = 61 + 14 = 75°
- ∠B = 7x + 1 = 7(10.18) + 1 ≈ 71.26 + 1 = 72.26°
- ∠C = given as 105°
- ∠D = 7y + 1 = 7(15.25) + 1 = 106.75 + 1 = 107.75°
Check the options:
- C) ∠A = 74° → False
- D) ∠B = 75° → False
- E) ∠C = 108° → False
None of the options match the calculated values exactly. However, ∠D (107.75°) and given ∠C (105°) are supplementary to angles very close to ∠A and ∠B, respectively.
Conclusion:
None of the options are exactly correct, but the closest correct statement is:
∠A = 75°, which makes option D very close, but not exact.
Correct answer: None of the given statements are entirely correct.
