The radius of the circle is 10 cm. The area of the triangle OAB is 600

Question

https://gmatclub.com/forum/download/file.php?id=62450 The radius of the circle is 10 cm. The area of the triangle OAB is 600 cm^2 . OA>OB. Find the coordinate of the point P. A. (15:20) B. (12:18) C. (16:18) D. (16:20) E. (15:18) triangle.jpg

IanStewart · Answer

The center of the circle is at (10, 10). Every point on the circle must be a distance of 10 from the center. Going from the center to answer choice C, (16, 18), we'd go right 6 units and up 8 units, so we'd get a 6-8-10 triangle, and answer choice C is on the circle. No other answer choice is, so that answer must be correct.

I didn't try to solve the question properly (i.e. without answer choices), but at a glance, it doesn't look like it would be pleasant, algebraically speaking.

I didn't try to solve the question properly (i.e. without answer choices), but at a glance, it doesn't look like it would be pleasant, algebraically speaking.

sherxon · Answer

You solved it the gmat way)

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raju1212 · Answer

Since the graph is at right angle the triangle should follow a triplets. Hence its length ate in ratio 3:4:5 3 and 4 being the base and height (3*4=12) we can confirm this by getting base * height from triangle area.
So now we know height is 30 and base is 40. 
Max height of circle on y axis is 20(diameter) and since the tangent is at angle height will be slightly less than 20 
Same analogy at x axis and we can deduce it is closer to center hence it should be slightly more than 10. So my guess is 
12 :18 (B)

Posted from my mobile device

Thelegend2631 · Answer

You were right until first part.

You might want to refer to  IanStewart  solution above.

Centre will be at 10,10
Distance from this point to any point given in the options would be 10
Only C satisfies the condition

Or else  
If things get trickier with different options

Since we know area and incentre.

Semi perimeter = Area/Incentre Radius

Simple

S= 600/10 = 60

Sum of sides = 120

So we know this could be 30,40,50 triplet.

If you were given multiple options that satisfies distance is 10 units.

Since we know intercepts of x and y axis by knowing the lengths of the legs
The hypotenuse will form

X/40 + Y/30 = 1

So only one such option or one such coordinate will pass through the above equation or satisfy the above condition

Here option C

16/40 + 18/ 30 = 1
1 = 1

Hence c

CEO2021 · Answer

IanStewart

but how to overcome any option ( 18,16) & (16,18)

thx

IanStewart · Answer

If you draw the picture, but draw line AB with a slope of -1, then since AB is a tangent line to the circle, it's perpendicular to the radius, so the radius that meets point P would then have a slope of 1 (the negative reciprocal of -1). But that radius also passes through the center of the circle, (10, 10), so the radius line would have the equation y = x, and the point P would then have equal x and y coordinates (the point would be (10 + 5√2, 10 + 5√2).) That's where the point would be if the slope of AB was -1, which is not the situation we have. If you then imagine moving the points A and B, moving B up and A to the left, so that line AB falls more steeply (but still touches the circle at one point), you'll see that the point P moves down and to the right, and its x-coordinate grows while its y-coordinate shrinks. If instead you move A to the right, and B down, so the slope of AB gets closer to zero, then point P moves up and to the left. So its x-coordinate gets smaller and its y-coordinate gets larger. That's what we're doing in this question, because OA > OB, so the x-coordinate of P must be less than the y-coordinate of P, and if we saw the options (18, 16) and (16, 18), we'd know (16, 18) was right. The answer (18, 16) would be right if instead OB > OA.

If instead you're asking how to solve without answer choices, then, if you're only using GMAT-level math, I don't think you can, or at least not without doing an enormous amount of work (unless you just make a lucky guess at the dimensions of the triangle). The solution hD13 outlined above looks like the fastest one to me, but it uses facts you'd never need on the test. When you inscribe a circle in a triangle, there is a relationship between the circle's radius and the perimeter and area of the triangle: (half the perimeter)*(radius) = Area. You would never need to know that on the GMAT, but using it here, we find the perimeter is 120. If the area is 600 and the perimeter is 120, there's only one triangle we can have (if you don't care which side is shortest), and with the perimeter and area of the triangle, you can either solve algebraically or see by inspection that we have a 30-40-50 triangle. Then there are a few options to finish the problem - for example, since we now have two points on the line, we can produce its equation. We also can produce an equation for the circle (though you really never need to do that on the GMAT for circles that are not centred at the origin) and just solve algebraically for their intersection point. Or with the equation of the line, you can then easily distinguish between answer choices if several answers lie on the circle.

Real GMAT questions aren't set up this way though, where the only convenient way to solve is by eliminating the answers that don't make sense (that aren't on the circle). In a real GMAT question, there will always be some practical two-minute way to arrive at the answer if you 'do the math'.

Thelegend2631 · Answer

Since we know area and incentre.

Semiperimeter=Area/IncentreRadiusSemiperimeter=Area/IncentreRadius

Simple

S= 600/10 = 60

Sum of sides = 120

So we know this could be 30,40,50 triplet.

If you were given multiple options that satisfies distance is 10 units.

Since we know intercepts of x and y axis by knowing the lengths of the legs
The hypotenuse will form

X/40+Y/30=1X/40+Y/30=1

So only one such option or one such coordinate will pass through the above equation or satisfy the above condition

Here option C

16/40 + 18/ 30 = 1
1 = 1

Hence c

The radius of the circle is 10 cm. The area of the triangle OAB is 600

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The radius of the circle is 10 cm. The area of the triangle OAB is 600

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards