9 votes 9 votes Five line segments of equal lengths, $\text{PR, PS, QS, QT}$ and $\text{RT}$ are used to form a star as shown in the figure above. The value of $\theta$, in degrees, is _______________ $36$ $45$ $72$ $108$ Quantitative Aptitude gatecivil-2021-set1 quantitative-aptitude geometry + – Arjun asked Feb 19, 2021 • recategorized Apr 11, 2021 by Lakshman Bhaiya Arjun 2.6k views answer comment Share Follow Migrated from GO Civil 2 years ago by Arjun See 1 comment See all 1 1 comment reply Goal commented May 10, 2022 reply Follow Share Your answer is correct but approach is wrong. Angle between S& R never be 2 theta. If it so then theta would be 45 degree. Consider triangle TQX then 4theta=180 & theta will be 45 degree. 0 votes 0 votes Please log in or register to add a comment.
6 votes 6 votes $\textbf{Central Angle Theorem:}$ The Central Angle Theorem states that the central angle from two chosen points $A$ and $B$ on the circle is always twice the inscribed angle from those two points. The inscribed angle can be defined by any point along the outer arc $AB$ and the two points $A$ and $B.$ Now, we can use the above theorem, and get the below diagram, Now, $10\theta = 360^{\circ}$ $\implies \theta = 36^{\circ}.$ So, the correct answer is $(A).$ Reference: http://jwilson.coe.uga.edu/EMAT6680Su09/Park/As8dspark/As8dspark.html Lakshman Bhaiya answered Mar 27, 2021 • reshown May 14, 2022 by Lakshman Bhaiya Lakshman Bhaiya comment Share Follow See 1 comment See all 1 1 comment reply KushaA commented Jul 1, 2021 reply Follow Share Provided link isn’t working.. Can you elaborate how you assumed intersection point of SQ and RT as origin to apply theorem. Instead one can observe that Pentagon is regular due to symmetry,as all line segments are equal; Hence each of its interior angles is 108deg, Using this we can arrive at Ѳ=36deg. 1 votes 1 votes Please log in or register to add a comment.
5 votes 5 votes Here if we get that its a regular pentagon then it becomes easy Zenitsu answered May 14, 2022 Zenitsu comment Share Follow See all 7 Comments See all 7 7 Comments reply shishir__roy commented May 14, 2022 reply Follow Share How did you prove that the Pentagon is regular? 1 votes 1 votes Zenitsu commented May 14, 2022 reply Follow Share Its a little long proof but it is done by proving the 5 isosceles triangles congruent so the base of each side is equal and to proof them equal external angle equals 2 opp interior angle and in a circle that 2theta property is used as u will see we can draw a circle with all vertex touching its circumference 0 votes 0 votes Arjun commented May 14, 2022 reply Follow Share Yes, that proof is needed to answer this question. There are 5 isosceles triangles. Rest needs to be proved. 0 votes 0 votes Zenitsu commented May 15, 2022 reply Follow Share I have proved 2 triangles like that all 5 can be proved congruent and because of that their bases are same and hence proves that its a regular pentagon 0 votes 0 votes Zenitsu commented May 15, 2022 reply Follow Share Sir if u say i will edit and add this image in main ans 0 votes 0 votes Arjun commented May 15, 2022 reply Follow Share Now, how did you prove that their bases are the same? :) 0 votes 0 votes Zenitsu commented May 16, 2022 reply Follow Share Parts of 2 congruent triangles are equal so by congruency i m able to say their bases must be equal 0 votes 0 votes Please log in or register to add a comment.