Triangle

[beater]

Please explain

[sureshbala]

Clearly the diameter is the value of a+b.



From right angle triangle PSQ, PQ = sqrt(4+a^2)

From right angle triangle QSR, QR = sqrt(4+b^2)

Also in triangle PQR, angle Q = 90 degrees since angle in a semi circle is right angle.

So considering the right angle triangle PQR, by Pythagoras theorem we have

PQ^2 + QR^2 = PR^2

i.e (4+a^2) + (4+b^2) = (a+b)^2

Solving this we get ab = 4.

So from the given question itself one can conclude that ab=4.

So from statement 1 since a =4, we can conclude b =1 (since ab=4) and hence diameter is 5

Similarly from statement 2 since b=1 we can conclude a =4 and hence diagmeter is 5.

Thus either statement alone is sufficient to answer this question.

Hence D will be the answer

[beater]

Anyone with a different approach?

Does the concept of similar triangles work in this question?

[Ian Stewart]

Yes, similar triangles is a good approach here. It's useful to know: if you connect a diameter of a circle to any other point on the circumference, you will have a 90 degree angle at that point. So angle PQR is 90. From here, if you label the remaining angles, you'll see that all three triangles are similar; the large triangle is just a magnified version of the smaller triangles.

From each statement we can work out all three sides in one triangle, so we know the ratio of the lengths of the sides. With that ratio, we can find the length of all three sides in any similar triangle, provided we know one side, so each statement is sufficient to find every length in the picture. That is, no calculation is required if you notice the triangles are similar.

[sureshbala]

Yes, we can also use similar triangles concept to answer this.

In fact you can answer this by considering only two triangles and there is no need to consider the bigger triangle here. Of course we should know that angle in a semi circle is 90 in order to conclude that the two smaller triangles are similar to each other.

Triangle PSQ is similar to QSR

So we have PS/SQ = QS/SR

i.e a/2 = 2/b

So ab = 4.

Hence either statement alone is sufficient

[canindia]

Nice definition Suresh

[gaggleofgirls]

I must be tired, because my first thought when I looked at the problem was that, at the very least, It is C, not E because the problems asks what is a + b/ 1) gives us a and 2) gives us b so certainly the combination gives us a + b

The similar triangles is clearly the right solution where knowing either a or b gets you a+b and the answer is D.

-Carrie

[mrsmarthi]

There is one more theorem - When an altitude is drawn on to the hypotenuse, then the product of the line segments on the hyptenuse = square of the height.

From our problem, a*b = 2 ^ 2 ==> ab = 4.

Knowing one of a / b will give the other value. So D will be sufficient.