If arc PQR above is a semicircle,

If arc PQR above is a semicircle, what is the length of diameter PR?

3 Explanations

▲

hichem elkateb

how those two triangles are similar , and why we do long leg over short leg ?

Dec 10, 2015 • Comment

Cydney Seigerman, Magoosh Tutor

How are the two triangles similar?
As Jonathan explained in an earlier post, similar triangles have congruent angles (the measures of their angles are the same). In this problem, we actually have three similar triangles: PQR, PTQ, and QTR. Because PR is the diameter, the inscribed angle PQR is 90?. Angles PTQ and TQR are also 90?. Angle P is shared by triangles PQR and PTQ. Because the triangles have to congruent angles, the measure of their third angle must also be equal, which means that angle PQT = angle R. Therefore, the three triangles have angles of the same measure. Mike shows which angles are equal in the video explanation.

Why do we do long leg over short leg?
We could have flipped the fractions and divided short/long to have gotten the same result:

short/long: 2/a = b/2 --> 4 = ab

What's important is that we are consistent in which leg is in the numerator and which is in the denominator.

I hope this helps! :)

Dec 14, 2015 • Reply

▲

Jonathan , Magoosh Tutor

Sure :) The first key is that we know angle PQR is 90 because it is an inscribed angle holding the diameter. We also know angle PTQ = angle QTR = 90.

Therefore, we can show the big/original triangle PQR is similar to each of the smaller triangles because the big triangle triangle shares two common angles with each triangle; therefore, the third angle must be the same as well; therefore, the triangles must all be similar.

For example, triangle PQR and triangle PQT both have angle TPQ and a 90 degree angle. Therefore, angle PQT must equal QRT.

So angle PQT = angle QRT and angle TPQ = angle TQR and all three triangles are similar.

Jan 19, 2015 • Comment

hichem elkateb

thank you , the similarity is clear but how about the fraction used to find a.b equal to 4? and which lesson should i see to understand that logic
Thank you

Dec 10, 2015 • Reply

Cydney Seigerman, Magoosh Tutor

To get these ratios, we need to know that corresponding sides of similar triangles have the same ratio. So, in this question, PT/TQ = QT/TR. PT = a, TR = b, and QT = TQ = 2. We can determine which sides are correspond to each other based on the angle measurements (corresponding sides are between congruent angles), which is discussed in more detail in this blog post (http://magoosh.com/gmat/2013/gmat-math-similar-shapes/) and our lesson video on similar triangles (http://gmat.magoosh.com/lessons/292-similar-triangles).
Hope this clears up your questions!

Dec 14, 2015 • Reply

▲

Mike McGarry, Magoosh Tutor