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CAT 2019 Slot 1 — QA Question 8

TrianglesEasy

Let T be the triangle formed by the straight line 3x + 5y − 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is

Answer & solution

Correct answer: 9

Answer: 9

Solution

Easy

The line meets the axes at two points which, with the origin, form a right triangle (the right angle is at the origin). For a right triangle the hypotenuse is the diameter of the circumcircle, so the radius is half the hypotenuse.

O C(15,0) A(0,9)
1

Find the intercepts. Set y=0y=0 then x=0x=0 in 3x+5y−45=03x+5y-45=0.

y=0: 3x=45⇒x=15⇒ C(15,0)x=0: 5y=45⇒y=9⇒ A(0,9)\begin{aligned} &y=0:\ 3x=45 \Rightarrow x=15 \quad\Rightarrow\ C(15,0)\\ &x=0:\ 5y=45 \Rightarrow y=9 \quad\Rightarrow\ A(0,9) \end{aligned}
2

Identify the right angle. The third vertex is the origin O(0,0)O(0,0), where the axes meet at 90∘90^\circ. So ∠O=90∘\angle O = 90^\circ and ACAC is the hypotenuse, hence the diameter of the circumcircle.

diameter=AC(hypotenuse of right triangle)\begin{aligned} &\text{diameter} = AC \quad\text{(hypotenuse of right triangle)} \end{aligned}
3

Compute the hypotenuse. Distance from A(0,9)A(0,9) to C(15,0)C(15,0).

AC=152+92=225+81=306\begin{aligned} &AC = \sqrt{15^2 + 9^2} = \sqrt{225+81} = \sqrt{306} \end{aligned}
4

Halve to get the radius.

L=3062≈17.492≈8.74\begin{aligned} &L = \frac{\sqrt{306}}{2} \approx \frac{17.49}{2} \approx 8.74 \end{aligned}

The nearest integer is 99.

integer closest to L=9\text{integer closest to } L = 9

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CAT 2019 Slot 1 QA Q8: Let T be the triangle formed by the straight line 3x + 5y − 45 = 0 and the coordinate axes. Let the circ — Solution | TheCATExam