GATE 2020 EE GA Q.9: solution and answer

GATE 2020 · EEGA Q.9 · 2 marks · MCQ

Given a semicircle with $O$ as the centre, as shown in the figure, the ratio $\frac{A C + C B}{A B}$ is ______, where $\overline{A C}$ , $\overline{C B}$ and $\overline{A B}$ are chords.

GATE 2020 EE GA question 9: A semicircle drawn with a thick black arc above a horizontal diameter AB. — Figure from the GATE 2020 EE paper. It is part of the question.

(A)
$2$
(B)
$3$
(C)
$2$
(D)
$3$

Answer and solution

ANSWER

The ratio $\frac{A C + C B}{A B} = 2$ , which is option (A).

Official answer: ASource: Official answer key, GATE 2020 EE (IIT Delhi)

Let the radius be $r$ . Then $O A = O B = O C = r$ , and the diameter is $\overline{A B} = 2 r$ .
The figure marks $O C ⊥ A B$ . So C is the top of the arc, and triangles $A O C$ and $B O C$ both have a right angle at $O$ .
Apply Pythagoras' theorem in $△ A O C$ : $\overline{A C} = O A^{2} + O C^{2} = r^{2} + r^{2} = r 2$ . The same working in $△ B O C$ gives $\overline{C B} = r 2$ .
Check: the angle in a semicircle is $9 0^{\circ}$ , so $∠ A C B = 9 0^{\circ}$ and $\overline{A C}^{2} + \overline{C B}^{2}$ should equal $\overline{A B}^{2}$ . Here $2 r^{2} + 2 r^{2} = 4 r^{2} = (2 r)^{2}$ , so the lengths agree.
Substitute into the ratio: $\frac{A C + C B}{A B} = \frac{r 2 + r 2}{2 r} = \frac{2 2 r}{2 r} = 2$
The radius cancels, so the ratio is $2$ for any semicircle. That is option (A).

KEY IDEA

When C is the top of the semicircle ( $O C ⊥ A B$ ), the triangles $A O C$ and $B O C$ are right-angled at $O$ , with both legs equal to the radius $r$ . So each chord is $r 2$ and the diameter is $2 r$ . The ratio does not depend on $r$ and equals $\frac{2 r 2}{2 r} = 2$ .

WATCH OUT FOR

One mistake is to take each chord as equal to the radius, which gives $\frac{2 r}{2 r} = 1$ . Another is to use $\overline{A B} = r$ in place of $2 r$ , which gives $22$ . A third is to add the chords as squares, since $\overline{A C}^{2} + \overline{C B}^{2} = \overline{A B}^{2}$ , and then pick option (C) $2$ . The question adds the lengths, not their squares.

From the GATE 2020 EE paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

GATE 2020 EE GA Q.9: solution and answer

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