zaarathelab wrote:In the diagram, Triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has length 12. PQ>QR. What is the ratio of area of triangle PQS and the area of triangle RQS.
1. 3/2
2.7/4
3. 15/8
4.16/9
5. 2
ALWAYS LOOK FOR SPECIAL TRIANGLES.
Since the perimeter of ∆PQR is an integer, ∆PQR is almost certainly a pythagorean triple: 3-4-5, 5-12-13, etc.
The perimeter of a 3:4:5 triangle = 3+4+5 = 12.
To yield a perimeter of 60, all the sides must be multiplied by a factor of 5:
15+20+25 = 60.
Since PQ>PR, ∆PQR likely looks like this:
To confirm that we have the correct dimensions for ∆PQR:
If we call QR the base and PQ the height, bh = 20*15 = 300.
Since QS = 12, if we call PR the base and QS the height, bh = 25*12 = 300.
Since bh is the same in each case, we have determined the correct dimensions of ∆PQR.
When a height is drawn through the right angle of a right triangle, all the resulting triangles are similar.
Thus, ∆PQS is similar to ∆RQS.
The hypotenuse of ∆PQS = 20.
The hypotenuse of ∆RQS = 15.
Thus, corresponding sides in ∆PQS and ∆RQS are in a ratio of 20:15 = 4:3.
Given similar triangles with corresponding sides in a ratio of x:y, the ratio of the areas = x² : y².
Thus, the ratio of the areas = 4² : 3² = 16:9.
The correct answer is
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